From 53e331ed6f56dac05ca084f992430924a36bce55 Mon Sep 17 00:00:00 2001 From: mfherbst Date: Thu, 21 Nov 2024 16:04:53 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20epfl-mat?= =?UTF-8?q?mat/error-control-modelling@280445edaeef694af71d7cb0f2df4862e82?= =?UTF-8?q?e4cd1=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 01_Introduction.html | 20 + 01_Introduction.jl | 769 ++++ 01_Introduction.plutostate | Bin 0 -> 92993 bytes 01_Presentation/.gitignore | 10 + 01_Presentation/about_you.tex | 102 + 01_Presentation/compile.sh | 6 + 01_Presentation/course.tex | 119 + 01_Presentation/img/disasters/Ariane_5.jpg | Bin 0 -> 395509 bytes .../img/disasters/Ariane_5.jpg.source | 1 + 01_Presentation/img/disasters/patriot.jpg | Bin 0 -> 5330 bytes .../img/disasters/patriot.jpg.source | 1 + 01_Presentation/img/disasters/sleipner.gif | Bin 0 -> 53878 bytes .../img/disasters/sleipner.gif.source | 1 + 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\ No newline at end of file diff --git a/01_Introduction.jl b/01_Introduction.jl new file mode 100644 index 0000000..6e0d29f --- /dev/null +++ b/01_Introduction.jl @@ -0,0 +1,769 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═║ 52b4d68c-d6f1-11ee-18b4-db386a5995ad +begin + using HypertextLiteral + using PlutoUI + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ b944f68e-85a6-4613-8c5e-0fe3d47ff0d4 +md""" +# Introduction +""" + +# ╔═║ 53e49738-8673-4ecd-9e86-b1431200b132 +md""" +## Opening remarks + +```math +\text{\textit{All models are wrong, only some models are useful.}} +``` + +**[Slides of the presentation](/)**: These summarise the following paragraphs and the content of the course in a short presentation. +$\textcolor{red}{\text{todo: add slide link here}}$ +""" + +# ╔═║ aa175689-f915-42a1-b472-19deb5811f46 +md""" +Constructing suitable **approximations to model the world** around us +sits at the **heart of all sciences**. As the above famous quote implies, +modelling a scientific problem inevitable comes with the introduction +of errors. Even the most accurate models of physics, e.g. the standard model +of particle physics or relativistic quantum mechanics have their limitations. +Moreover these models are typically too involved to be applied to +all but the simplest systems. In particular for "real" systems of macroscopic +size additional approximations. +Typically this not only involves the selection of **more approximate**, +thus simpler, +**physical models**, but also employing **numerical** instead of an analytical techniques +for solving the underlying equations. +""" + +# ╔═║ 7229e2b6-cf77-4b32-b52d-1fc7d218b007 +md""" +Which physical effects can be neglected and which numerical techniques are most +appropriate is often application-specific and **many alternatives exist**. +Understanding the **dominant sources of error** in a simulation is thus +a crucial component in understanding whether each respective **approximation choice** has been appropriate and --- most importantly --- which of them **needs to be refined** in order to improve the result. +""" + +# ╔═║ bd045f8e-7baa-4fbf-94bc-22e39cac9a0d +md""" +The goal of this lecture is to discuss mathematical techniques +which aid with the estimation of error in scientific simulations. +Within the lecture we will distinguish various forms of error and consider (some) of them in separate discussions throughout the semester. For example we will distinguish the **model error**, i.e. the error introduced the selected physical model compared to the ground truth, +as well as the **numerical error**, +i.e. all additional error from the numerical procedure employed for +solving the chosen physical model. +Of particular focus in the lecture will be understanding the various aspects +of the numerical error, +that is in particular the errors due to the chosen **discretisation basis**, +the **algorithms** to solve the numerical problem as well as the chosen +**floating-point arithmetic**. +""" + +# ╔═║ cd9950c8-1b3d-4a1b-8606-9fcf4f39599f +md""" +Looking back into the **history of scientific modelling** +a number of incidents with **hundreds of millions of dollars +of damage** could have been **prevented** if better numerical error control +measures would have been implemented. +One example is the +[sinking of the Sleipner A oil rig](https://en.wikipedia.org/wiki/Sleipner_A) +platform +due to an insufficient finite element discretisation. +A second example is the +[self-destruction of an Ariane 5 rocket](https://en.wikipedia.org/wiki/Ariane_5) +in its test flight +due to an error in the performed floating-point arithmetic. +In this lecture we will discuss techniques how such errors could have been prevented. +Some guiding questions for our lecture are: +- What can go wrong in a simulation? +- How can we detect this and what can we do about it? +- If we know errors how can this be used to make simulations faster and more reliable? +""" + +# ╔═║ d6db19e2-78fb-4619-b2c7-c2784461c583 +md""" +## Eigenvalue problems in scientific modelling + +Eigenvalue problems occur naturally in many domains of science. +In physics and engineering problems they are frequently related to vibrations, +e.g. bridges or buildings can swing under wind load. +A famous example is the Tacoma narrows bridge, which collapsed in 1940 just half +a year after its opening due to strong winds bringing the bridge into strong vibrations. +Similarly the London millennium foot bridge had to be closed due to its strong wobbeling +by the passing pedestrians. + +In my own field, atomistic modelling, the underlying workhorse theory is the SchrΓΆdinger equation. +In quantum mechanics the eigenpairs (states) of an operator describing the physical system under +study form the key quantity of interest in simulations. + +Moreover, as we will discuss in the next examples in more detail, +there is a natural connections from solving partial +differential equations~(PDE) to eigenvalue problems involving the PDE operator. + +Another natural connection arises when analysing the convergence of +iterative numerical schemes, +such as multidimensional fixed-point problems, +iterative schemes like Newton or the conjugate-gradient algorithm. +The convergence of CG, for example, +is related to the condition number of the underlying linear system, +which in turn are related to singular values of the system matrix +--- an Quantity, which can be obtained by solving an eigenvalue problem. +""" + +# ╔═║ 189b99a5-776f-4ad6-b05a-66b9801a97c4 +md""" +## The connection between linear and eigenvalue problems : the heat equation + +Let $\Omega \subset \mathbb R^n$ with a Lipshitz (smooth) boundary. +We seek $u : \Omega \times \mathbb R^+ β†’ \mathbb R$ such that +```math +\begin{align} +\frac{βˆ‚ u}{βˆ‚ t} - \laplacian u &= 0 && \text{in } \Omega \times \mathbb R^+ \tag{Heat equation} +\\ +u(\cdot,0) &= u_0 && \text{in } \Omega \tag{Initial conditions} +\\ +u &= 0 && \text{on } βˆ‚ \Omega \times \mathbb R^+ \tag{Dirichlet b.c.} +\end{align} +``` +Where the initial conditions $u_0 : \Omega β†’ \mathbb R$ are of appropriate regularity. + +""" + +# ╔═║ e52eb9de-c5b3-424e-b2d5-e1b926f76fd4 +md""" +To solve this problem, we perform separation of variables +```math +u(x,t) = \sum_{n = 1}^{\infty} c_n(t) v_n(x). +``` +We will see later that this postulate is justified. + +To start, let's just consider one term of the sum, i.e. $u(x,t) = c(t) v(x)$. + +- From the heat equation + ```math + c'(t) v(x) - c(t) \laplacian v(x) = 0 + ``` + +- Assuming that $v(x)$ and $c(t)$ are non-zero, we can divide + ```math + \begin{align} + \frac{c'(t)}{c(t)} = \frac{\laplacian v(x)}{v(x)} && βˆ€ \ t>0, x ∈ \Omega. + \end{align} + ``` +- The left only depends on $t$, and the right only on $x$. + As a result, we need + ```math + \frac{c'(t)}{c(t)} = \frac{\laplacian v(x)}{v(x)} = ΞΌ \in \mathbb R + ``` + for some $ΞΌ$. +""" + +# ╔═║ d5581136-7b5e-4987-a67d-241286d424c8 +md""" +- Consider the time-dependant part ($c$) + ```math + \begin{align} + c'(t) &= ΞΌ c(t) + \\ + \Rightarrow c(t) &= Ξ± \exp(ΞΌ t). + \end{align} + ``` +- Separately, consider the space-dependant part ($v$) + ```math + \begin{align} + \laplacian v &= ΞΌ v && \text{in } \Omega + \\ + v &= 0 && \text{on } βˆ‚\Omega + \end{align} + ``` + *which is the Dirchlet-Laplace eigenvalue problem* ! +""" + +# ╔═║ 4db1319a-0b7c-42d5-ba03-00f106542c80 +md""" +- One can show that the operator $- \laplacian$ with homogeneous Dirichlet boundary conditions has a sequence of eigenvalues $0 < Ξ»_1 ≀ Ξ»_2 ≀ … β†’ ∞$ and orthonormal eigenfunctions $\{ \phi_n \}_{n=1}^∞$, i.e. + ```math + - \laplacian \phi_n = Ξ»_n \phi_n + ``` + with + ```math + \langle \phi_n, \phi_m \rangle = ∫_\Omega \phi_n(x) \phi_m(x) dx = Ξ΄_{mn}. + ``` + +- These form a complete basis of $H^1_0 (\Omega)$, the relevant Hilbert space for this problem. + +This justifies the ansatz +```math +u(x,t) = \sum_{n=1}^∞ c_n(t) \phi_n(x) +``` +we made before, where $\phi_n$ are now eigenfunctions of the Laplace operator. +""" + +# ╔═║ 17dd79e4-c998-48cc-854e-34e20f3338f9 +md""" +To fully translate the problem to the eigenfunction setting, we need to consider the initial condition $u_0.$ + +- On the one hand, in the eigenbasis, + ```math + u_0 = \sum_{i=1}^∞ \underbrace{\langle \phi_i, u_0 \rangle}_{u_{0,i}} \phi_i. + ``` + +- On the other hand, + ```math + c_i(t) = Ξ±_i \exp(- Ξ»_i t). + ``` + which yields + ```math + u(x,t) = \sum_{i=1}^∞ Ξ±_i \exp(- Ξ»_i t) \phi_i(x). + ``` + +- Taking $t β†’ 0$, we obtain + ```math + \sum_{i=1}^\infty u_{0,i} \phi_i(x) = u_0(x) = u(0,x) = \sum_{i=1}^\infty \alpha_i \phi_i(x). + ``` + +- Since $\phi_i$ is a basis, we deduce $\alpha_i = u_{0,i}$ and finally obtain + ```math + u(x,t) = \sum_{i=1}^\infty u_{0,i} \exp(- \lambda_i t) \phi_i(x) + ``` + +We observe the following : +- The solution decays to $u_\infty = 0$ as $t \to \infty$. +- The decay rate depends on $\lambda_i$. +""" + +# ╔═║ 9f07f419-fbe9-4ff5-8cbb-2bd056868b61 +md""" +More importantly for this course, this shows that eigenvalue problems, which we will focus on, are key in understanding linear problems, even though they might seem unrelated at a first glance. +One of these linear problems of particular relevance is solving SchrΓΆdinger equation, which we treat in the next section. +""" + +# ╔═║ b102f351-f5b1-47c0-8737-f813b5b745f6 +md""" +## The SchrΓΆdinger equation and quantum mechanics + +At the microscopic level, the physics and chemistry of materials is governed by the interaction of electrons and nuclei. +At this scale, the regime of quantum mechanics applies. + +In quantum mechanics the state of a system is described the complex-valued square-integrable wave function $\Psi : (x,t) ↦ \Psi(x,t) \in \mathbb C$, where $x$ corresponds to the degrees of freedom (e.g. position or spin) of the particles (e.g. electrons) in the system. + +For simplicity, take $x \in \mathbb R^3$, i.e. one particle in 3D. +Part of the meaning of $\Psi$ appears through its modulus squared. +Indeed, for all $t$, $|\Psi(x,t)|^2$ corresponds to the probability distribution of finding the particle at position $x$ at time $t$. + +Further information appears through its Fourier transform, as $|\hat \Psi (p,t)|^2$ corresponds to the probability distribution of finding the particle at momentum $p$ at time $t$, where the Fourier transform is given by +```math +\hat \Psi (p,t) = \int_{\mathbb R^3} \Psi (x,t) e^{-2 \pi i \ p \cdot x}dx. +``` +For those already familiar with quantum mechanics, note that we set $\hbar = 1$. +""" + +# ╔═║ e1132fa4-c9ff-487e-b716-0b3157c2cdd0 +md""" +In analogy to classical mechanics, one may obtain the dynamics of the particle by investigating its total energy (considering the mass of the particle to be 1). + +- The total energy is given by + ```math + \begin{align} + E(t) &= \text{kinetic energy} + \text{potential energy} + \\ + &= \int_{\mathbb R^3} \frac{|p|^2}{2} | \hat \Psi (p,t) |^2 dp + \int_{\mathbb R^3} V(x) | \Psi (x,t) |^2 dx + \end{align} + ``` + where $V : \mathbb R^2 \to \mathbb R$ is the potential inducing the particle dynamics. + +- From the Fourier identity $p \hat \Psi (p) = -i \widehat{(\nabla \Psi)}(p)$, we can develop the first term into + ```math + \begin{align} + \int_{\mathbb R^3} |p|^2 | \hat \Psi (p,t) |^2 dp &= \int_{\mathbb R^3} |p \hat \Psi (p,t)|^2 dp + \\ + &= \int_{\mathbb R^3} \lvert -i \widehat{(\nabla \Psi)}(p,t)|^2 dp + \\ + &= \int_{\mathbb R^3} \lvert \widehat{(\nabla \Psi)}(p,t)|^2 dp + \\ + &\hspace{-0.9em} \stackrel{\text{Parceval}}{=} \int_{\mathbb R^3} | \nabla \Psi(x,t) |^2 dx + \end{align} + ``` + where we used Parceval's theorem to go from the third to the fourth line. +""" + +# ╔═║ 33c563af-3436-4275-becd-5333d480b4f4 +md""" +- With this we obtain, + ```math + \begin{align} + E(t) &= \frac1{2} \int_{\mathbb R^3} | \nabla \Psi(x,t) |^2 dx + \int_{\mathbb R^3} V(x) |\Psi(x,t)|^2 dx + \\ + & \stackrel{IBP}{=} - \frac1{2} \int_{\mathbb R^3} \overline{\Psi(x,t)} \laplacian \Psi(x,t) dx + \int_{\mathbb R^3} \overline{\Psi(x,t)} V(x) \Psi(x,t) dx + \\ + &= \int_{\mathbb R^3} \overline{\Psi(x,t)} \ \opH \ \Psi(x,t) dx + \\ + &= \langle \Psi, \opH \Psi \rangle_{L^2(\mathbb R^3)} + \end{align} + ``` + Where we used integration by parts to go from the first to the second line. + +- In this, the *Hamiltonian* + ```math + \opH \coloneqq - \frac1{2} \laplacian + V(x) + ``` + gives the total energy of the system when applied to the wavefunction $\Psi$, where + - $- \laplacian / 2$ is the kinetic energy operator + - $V(x)$ is the potential energy, applied point-wise in real space. +""" + +# ╔═║ 8e7daba1-277b-4632-b995-5be76616d0d3 +md""" +The dynamics of the system are now given by the time-dependant SchrΓΆdinger equation (TDSE) +```math +i \frac{\partial \Psi}{\partial t} = \opH \Psi \tag{TDSE} +``` +In general, solving this equation is quite complicated. +However, since it is linear, we know that superpositions of multiple solutions are also a solution. +This motivates a separation of variables similar to the heat equation. + +- Inserting $\Psi(x,t) = c(t) \varphi(x)$ into the TDSE yields + ```math + i \frac{c'(t)}{c(t)} = \frac{\opH \varphi(x)}{\varphi(x)} = E = \text{cst} + ``` + +- For the time-dependence we obtain the ODE + ```math + \begin{align} + c'(t) &= - i E c(t) + \\ + \Rightarrow c(t) &= \exp(- i Et) + \end{align} + ``` +- The space dependence yields the time-independent SchrΓΆdinger equation (TISE) + ```math + \opH \varphi(x) = E \varphi(x) \tag{TISE} + ``` +""" + +# ╔═║ 94c2368c-243e-4e29-ba9b-a6c9253fa74f +md""" +- As before, if we are able to find (orthonormal) eigenpairs $(E_n,\phi_n)$, then the TDSE can be solved as + ```math + \Psi(x,t) = \sum_{n =1}^\infty c_n e^{- i E_n t} \phi_n(x) + ``` + with $c_n = \langle \phi_n , \Psi \rangle$. + +- Of key importance in quantum mechanics in particular is the computation of the eigenpair corresponding to the smallest eigenvalue of $\opH$, called the *ground state*. +""" + +# ╔═║ d26799e2-bd9a-4ff5-89b2-7d4a8cc61b74 +TableOfContents() + +# ╔═║ 8993ae47-1173-46c1-8f1e-55412215b61f +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 265) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] 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"4536629a-c528-5b80-bd46-f80d51c5b363" +version = "0.3.23+4" + +[[deps.OrderedCollections]] +git-tree-sha1 = "dfdf5519f235516220579f949664f1bf44e741c5" +uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d" +version = "1.6.3" + +[[deps.Parsers]] +deps = ["Dates", "PrecompileTools", "UUIDs"] +git-tree-sha1 = "8489905bcdbcfac64d1daa51ca07c0d8f0283821" +uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" +version = "2.8.1" + +[[deps.Pkg]] +deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] +uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" +version = "1.10.0" + +[[deps.PlutoHooks]] +deps = ["InteractiveUtils", "Markdown", "UUIDs"] +git-tree-sha1 = "072cdf20c9b0507fdd977d7d246d90030609674b" +uuid = "0ff47ea0-7a50-410d-8455-4348d5de0774" +version = "0.0.5" + +[[deps.PlutoLinks]] +deps = ["FileWatching", "InteractiveUtils", "Markdown", "PlutoHooks", "Revise", "UUIDs"] 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+/presentation.nav +/presentation.log +/presentation.aux +/presentation.out +/presentation.vrb +/presentation.pdf +/presentation.snm +/presentation.toc +.*.swp +*~ diff --git a/01_Presentation/about_you.tex b/01_Presentation/about_you.tex new file mode 100644 index 0000000..55c4769 --- /dev/null +++ b/01_Presentation/about_you.tex @@ -0,0 +1,102 @@ +\begin{frame}{Your background \& prior knowledge} + \begin{center} + \LARGE{Your background and prior knowledge} + \end{center} +\end{frame} + +\begin{frame}{Vector spaces} + \noindent + Which of the following things is a \alert{vector space} over $\R$ + \vspace{1em} + \begin{enumerate} + \item $\R^n = \{(x_1, \ldots, x_n)^T , x_i \in \R \}$ + \item $\mathcal{F}(D, \R) = \{f : D \to \R\}$, + the set of all functions from $D$ to $\R$. + \item $\R^{n\times n}$: The set of all matrices + \end{enumerate} + \vspace{1em} + \visible<2>{\alert{Answer:} All of them!} +\end{frame} + +\begin{frame}{Inner products} + \noindent + Given vectors $x,y,z$ from an $\C$-vector space $V$. + What makes an \alert{inner product}? + \visible<2>{ + \vspace{1em} + \begin{itemize} + \item $\langle x, x \rangle \geq 0$ + \item $\langle x, y \rangle = \overline{\langle y, x \rangle}$ + \item $\langle x, \alpha y + \beta z \rangle + = \alpha \langle x, y \rangle + + \beta \langle x, z \rangle$ + \end{itemize} + \vspace{1em} + Examples: + \begin{itemize} + \item + $\langle x, y \rangle = x^H y = \overline{x^T} y = \sum_{x=1}^n \overline{x_i} y_i$ for vectors $x,y \in \C^n$ + \item + $\langle A, B \rangle_F = A^H B = \tr(A^H B)$ + for matrices $A,B \in \C^{n\times n}$ + \end{itemize} + } +\end{frame} + +\begin{frame}{Norms} + \noindent + Which of these statements is true: + \vspace{1em} + \begin{enumerate} + \item If $\langle x, y\rangle$ is an inner product, + then $\norm{x} = \sqrt{\langle x, x \rangle}$ is a norm + \item If $\langle x, y\rangle$ is an inner product + and $\norm{x} = \sqrt{\langle x, x \rangle}$, then + \[ \langle x, y \rangle < \norm{x} \norm{y} \] + \item If $\norm{x}$ is a norm, there exists an inner product + $\langle x, y\rangle$, such that $\norm{x} = \sqrt{\langle x, x \rangle}$ + \item Every norm satisfies the triangle inequality + \[ \norm{x + y} \leq \norm{x} + \norm{y} \] + \end{enumerate} + \vspace{1em} + \visible<2>{\alert{Answer:} 1 and 4 are true, 2 and 3 are false. + 2 is almost true the correct version is + \[ \langle x, y \rangle \leq \norm{x} \norm{y} \qquad \text{Cauchy-Schwarz}\] + } +\end{frame} + +\begin{frame}{Diagonalisation algorithms} + \noindent Which iterative diagonalisation algorithms do you know?\\ + \vspace{1em} + \noindent Who has studied before: + \vspace{1em} + \begin{itemize} + \visible<2->{\item Power method} + \visible<3->{\item Inverse power method} + \visible<4->{\item Rayleigh-quotient iteration} + \visible<5->{\item LOPCG} + \end{itemize} +\end{frame} + + + +% Sobolev spaces +% Hilbert spaces +% Numerical analysis +% PDE discretisations +% a posteriori error +% spectral theorem +% operators +% spectral theory +% diagonalisation algorithms +% Quantum mechanics +% Harmonic oscillator +% Plane waves +% Programming +% Julia + +\begin{frame}{I need a volunteer} + \begin{itemize} + \item \textbf{Your job:} Take pictures of blackboards + \end{itemize} +\end{frame} diff --git a/01_Presentation/compile.sh b/01_Presentation/compile.sh new file mode 100755 index 0000000..1e62bc9 --- /dev/null +++ b/01_Presentation/compile.sh @@ -0,0 +1,6 @@ +#!/bin/sh + +pdflatex presentation.tex && \ + pdflatex presentation.tex && \ + pdflatex presentation.tex && \ + rm presentation.log presentation.aux presentation.nav presentation.out presentation.snm presentation.toc diff --git a/01_Presentation/course.tex b/01_Presentation/course.tex new file mode 100644 index 0000000..910227d --- /dev/null +++ b/01_Presentation/course.tex @@ -0,0 +1,119 @@ +\begin{frame}{Structure of the course} + \begin{itemize} + \item \alert{Moodle:} \url{https://go.epfl.ch/error-control} + \vspace{-0.3em} + \item \alert{ED Forum:} Announcements \& \alert{student-driven} discussion + \vspace{1.0em} + \item Lectures split into two rough segments + \begin{itemize} + \vspace{-0.3em} + \item \alert{First half:} Matrix eigenvalue problems \& floating-point error + \vspace{-0.3em} + \item \alert{Second half:} Operator theory \& discretisation error + \vspace{-0.3em} + \item \alert{Notes:} \url{https://teaching.matmat.org/error-control} + \end{itemize} + \vspace{1.0em} + \item Attendance of exercises is \alert{expected} \textcolor{grey5}{(introduces new material!)} + \begin{itemize} + \vspace{-0.3em} + \item Discussion follows weekly exercise sheet + \end{itemize} + \end{itemize} +\end{frame} +\begin{frame}{Course evaluation} + \begin{itemize} + \item Marked semester project \textcolor{grey5}{($1/3$ of grade)} + \item Project interview \& oral exam \textcolor{grey5}{($2/3$ of grade)} + \item Project done in \alert{teams of $2-3$ students}. + \item Interdisciplinary teams are highly recommended + \vspace{1.5em} + \item Working on the projects + \alert{requires substantial time outside class} + \begin{itemize} + \vspace{-0.3em} + \item[$\Rightarrow$] We will setup \alert{survey} in week 2 to \alert{aid formation of groups} + \end{itemize} + \end{itemize} +\end{frame} + +\begin{frame}{Details on the exercises \& problem sheets} + \begin{itemize} + \item One problem sheet per week from moodle: \url{https://go.epfl.ch/error-control} + \vspace{1em} + \item Handing in of exercise sheets is optional + \begin{itemize} + \vspace{-0.3em} + \item Submission can only be done in the project group + \vspace{-0.3em} + \item Tutors give feedback on submitted sheets + \end{itemize} + \vspace{1em} + \item Initial exercises classes will be denser + \item Later exercise classes provide time to work on the project + \vspace{1em} + \item \alert{Important:} + \begin{itemize} + \vspace{-0.3em} + \item \textbf{Bring your laptop} to the exercise sessions + \vspace{-0.3em} + \item \textbf{Install \julia} programming language \textbf{beforehand} + \vspace{-0.3em} + \item[$\Rightarrow$] See \alert{instructions on moodle} + \end{itemize} + \end{itemize} +\end{frame} + +\begin{frame}{Details on the project} + \begin{itemize} + \vspace{-0.3em} + \item Project is essentially a larger problem sheet + \vspace{-0.3em} + \item One joint solution is submitted by each group + \vspace{-0.3em} + \item Responsibilities should be shared equally. + \vspace{1.0em} + \item During the oral exam \textcolor{grey5}{(about half the time)} + \begin{itemize} + \vspace{-0.3em} + \item Presentation of the problem sheet by student + \vspace{-0.3em} + \item Targeted follow-up questions + \vspace{-0.3em} + \item \textcolor{grey5}{(The other half of the oral + is about general course understanding)} + \end{itemize} + \vspace{1.0em} + \item Project evaluation criteria: + \begin{itemize} + \vspace{-0.3em} + \item See document on moodle + \end{itemize} + \vspace{1.0em} + \item Each group member obtains an \alert{individual mark}. + \end{itemize} +\end{frame} + +\begin{frame}{Topic of the semester project} + \begin{itemize} + \item Topic: \alert{Band structures with guaranteed error bars} + \begin{itemize} + \vspace{-0.5em} + \item Handout: 21st October \textcolor{grey5}{(tentative)} + \vspace{-0.5em} + \item Duration: 2 months, i.e.~projected deadline: \textbf{20th December} + \end{itemize} + \vspace{1.5em} + \begin{center} + \includegraphics[width=0.7\textwidth]{img/si_band_errors.pdf} + \end{center} + \vspace{0.5em} + \item Let's have a brief look at last year's projects \ldots + \end{itemize} +\end{frame} + +\begin{frame}{Questions ?} + \begin{center} + \huge{Questions ?} + \end{center} +\end{frame} diff --git a/01_Presentation/img/disasters/Ariane_5.jpg b/01_Presentation/img/disasters/Ariane_5.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d9f2f8083d739520c52256126d9003e9506ce552 GIT binary patch literal 395509 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Rinke, Patrick}, + doi = {10.1002/advs.201900808}, + issue = {21}, + journal = {Advanced Science}, + language = {en}, + month = {11}, + pages = {1900808}, + publisher = {Wiley}, + title = {Data-Driven Materials Science: Status, Challenges, and Perspectives}, + type = {article}, + url = {http://dx.doi.org/10.1002/advs.201900808}, + volume = {6}, + year = {2019}, +} diff --git a/01_Presentation/img/julia.pdf b/01_Presentation/img/julia.pdf new file mode 100644 index 0000000000000000000000000000000000000000..83d59b4f74543d6a3dbb34ce0d5a7e322becc39d GIT binary patch literal 3023 zcmZ8jc|4S999P-JluG3oEU&A`G53rs(HPfM7+0d*=6Eq;Ml(Z`q=VSV)fPD-gL3Ce zS`jN(LsUx2kqRpjgM_d%I=0>C^M0Q9eZJ58{C>ah@B94m)pD@3(nlCzAX>M^ix(ki z01hxIz7SJW0JaxocyoLJj1VG0004kl(fv3eOZfI9b3jXw%A|o1GcyRA!ve|v5N=xF z6@IMdZ5hJcyzJ*Q`=^(l`)Zh7Ydg`kk#pP{U&tmc=#Ux1Ge-4vl(!I}Eth8-bGRRSWlEtD)!CBnW6eEyj!v3d z0C&!NwxRK8Wo1s-koKP7h5nAVye&@q3@XcqkuguN4?PXFNjMl#x_7{M%P^%W7&TxV 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\begin{tikzpicture} + \node at (0, 0) {\includegraphics[width=\textwidth]{img/disasters/sleipner.png}}; + \only<2->{\node [fill=white,rotate=40] at (0, 0) {\alert{Broken and sunken}};} + \end{tikzpicture} + \end{center} + \visible<2->{ + \begin{itemize} + \item August 1991 + \item Sleipner A offshore platform + \item 1 billion dollar + \visible<3->{ + \item \alert{Too crude discretisation} + } + \end{itemize} + } + \vspace{0.7em} + \end{block} + \end{column} + + \begin{column}{0.3\textwidth} + \begin{block}{Intercept a missile} + \begin{tikzpicture} + \node at (0, 0) { + \includegraphics[width=\textwidth]{img/disasters/patriot.jpg} + }; + \only<2->{\node [fill=white,rotate=40] at (0, 0) {\alert{Missed Iranian missile}};} + \end{tikzpicture} + \visible<2->{ + \vspace{-1.0em} + \begin{itemize} + \item February 1991 + \item Patriot missile failure + \item 28 soldiers killed, 100 injured + \visible<3->{ + \item \alert{floating point conversion error} + } + \end{itemize} + } + \end{block} + \end{column} + \end{columns} + + \visible<3>{ + \vspace{1.0em} + \begin{itemize} + \smaller[2] + \item See website of Douglas N. Arnold for more details: + \url{https://www-users.cse.umn.edu/~arnold/disasters/disasters.html} + \end{itemize} + } +\end{frame} + +\begin{frame}{Ok, so these are the extreme cases, right?} + \begin{center} + \larger + \alert{Brainstorming:} Sources of error in scientific simulations + \end{center} + \vspace{3.0em} + \visible<2>{ + \begin{itemize} + \item Model + \item Numerics \textcolor{grey5}{\smaller (discretisation / basis set, algorithm, arithmetic)} + \item Implementation + \item Hardware \textcolor{grey5}{\smaller (CPUs have bugs!)} + \end{itemize} + } +\end{frame} + +\begin{frame}{Motivation in the \matmat group} + \begin{itemize} + \item \alert{21st century challenges}: + \begin{itemize} + \vspace{-0.3em} + \item Renewable energy, green chemistry, health care \ldots + \end{itemize} + \vspace{0.2em} + \item Current solutions limited by properties of available materials + \begin{itemize} + \vspace{-0.3em} + \item[$\Rightarrow$] Innovation driven by \alert{discovering new materials} + \end{itemize} + \vspace{0.2em} + \item Crucial tool: \alert{Computational materials discovery} + \begin{itemize} + \vspace{-0.3em} + \item Systematic simulations on \alert{$\simeq 10^4 - 10^6$ compounds} + \vspace{-0.3em} + \item Complemented by data-driven approaches + \vspace{-0.3em} + \item \alert{Noteworthy share} of world's supercomputing resources + \end{itemize} + \end{itemize} + \vspace{0.2em} + \definecolor{pgray}{rgb}{0.169,0.169,0.251} + \begin{center} + \includegraphics[height=3.3cm]{img/intro/roadmap-growth.png} + \hspace{1.5em} + \includegraphics[height=3.3cm]{img/intro/archer_usage.pdf} + \end{center} + \vspace{-1.3em} + \rule{4.5cm}{0.5pt}\\[-0.5em] + {\tiny \href{http://dx.doi.org/10.1088/1361-6463/aad926}{% + K. Alberi \textit{et. al.} J. Phys. D, \textbf{52}, 013001 (2019).}} +\end{frame} + +\begin{frame}{Sketch of high-throughput workflows} + \widenframe[4em]{ + \vspace{-0.2em} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{lpic}[r(2.5cm)]{./img/intro/funnel.png(.125)} + \lbl[l]{232,153;$\Big\}$\smaller[2.5] DFT PBE stability} + \lbl[l]{203, 95;\smaller[2.5]DFT PBE band gap} + \lbl[l]{187, 60;\smaller[2.5]Hybrid-DFT band gap} + \lbl[l]{175, 25;\smaller[2.5]Beyond DFT} + \end{lpic}\\[-0.7em] + {\smaller[2.5] Design funnel for photovoltaic materials} + \end{column} + \begin{column}{0.55\textwidth} + \includegraphics[width=5.0cm]{./img/intro/workflow_edit_parameters.png}% + \\[-0.5em] + {\smaller[2.5] Workflow for computing elasticity tensors} + \end{column} + \end{columns} + } + \vspace{-0.1em} + \begin{itemize} + \item Many parameters to choose \textcolor{grey5}{(algorithms, tolerances, models)} + \begin{itemize} + \vspace{-0.4em} + \item Elaborate heuristics: \alert{Failure rate $\simeq 1\%$} + \vspace{-0.4em} + \item Still: \alert{Thousands} of failed calculations + \vspace{-0.4em} + \item[$\Rightarrow$] \alert{Wasted resources} \& increased human attention + \textcolor{grey5}{\smaller (limits througput)} + \end{itemize} + % \vspace{0.1em} + % \item Carbon footprint? More complex design spaces? + \vspace{0.2em} + \item \textbf{Goal} in \matmat group: \alert{Self-adapting black-box algorithms} + \begin{itemize} + \vspace{-0.4em} + \item Transform \alert{empirical wisdom} to built-in \alert{convergence guarantees} + \vspace{-0.4em} + \item Requires: Uncertainty quantification \& error estimation + \vspace{-0.4em} + \item[$\Rightarrow$] Understand \alert{where and how} to spend efforts best + \end{itemize} + \end{itemize} + \vspace{-0.8em} + \rule{4.5cm}{0.5pt}\\[-0.5em] + {\tiny G.~Hautier Comput. Mater. Sci. \textbf{164}, 108 (2019); + L.~Himanen \textit{et. al.} Adv. Science \textbf{6}, 1900808 (2019).} +\end{frame} + +\begin{frame}{Broader vision: Robust \& error-controlled simulations} +\begin{itemize} + \item Error control $=$ \alert{Track simulation uncertainties}: + \begin{itemize} + \vspace{-0.30em} + \item Self-adapting simulations with mathematical guarantees + \vspace{-0.30em} + \item Integrate with error propagation efforts for surrogates% + \footnote{F.~Musil, A.~Grisafi \emph{et. al.} J. Chem. Theo. Comput. \textbf{15}, 2 (2019).} + \vspace{-0.30em} + \item [$\Rightarrow$] Byproducts: Data quality control, + accelerated design% + % \footnote{G.~Houchins and V.~Viswanathan MRS Bulletin \textbf{44}, 204 (2019).} + \end{itemize} + + \vspace{0.3em} + \item Error control $=$ \alert{Learn missing physics}: + \begin{itemize} + \vspace{-0.30em} + \item Data-enhanced models, active learning + \vspace{-0.30em} + \item Integration with experiment {\color{grey5} (autonomous discovery)} + \vspace{-0.30em} + \item [$\Rightarrow$] Exploit high-fidelity experimental, beyond-DFT data + \end{itemize} + + \vspace{0.3em} + \item Error control $=$ \alert{Leverage inexactness}: + \begin{itemize} + \vspace{-0.30em} + \item Error balancing: Optimal adaptive parameter selection + \vspace{-0.30em} + \item Randomised methods, selective precision {\color{grey5} (16-bit, FPGA)} + % Error control and multi-fidelity approaches pretty much unexplored + % in practical setting + \vspace{-0.30em} + \item Multi-fidelity approaches {\color{grey5} (reduced basis, surrogates)} + \end{itemize} + + \vspace{0.3em} + \item[$\Rightarrow$] Understand \alert{where and how} to spend efforts best + \vspace{-0.3em} + \item[$\Rightarrow$] Realm of mathematical research +\end{itemize} +\vspace{1em} +\end{frame} + + +\begin{frame}{Questions ?} + \begin{center} + \huge{Questions ?} + \end{center} +\end{frame} + +\begin{frame}{Focus of the course: Eigenvalue problems} + \begin{itemize} + \item Eigenvalue problems are ubiquitous, e.g. + \vspace{2em} + \item \alert{Vibrations of structures} + \begin{itemize} + \item Tacoma narrows bridge collapse 1940 + \item London millennium bridge construction flaw + \end{itemize} + \vspace{1em} + \item \alert{Quantum states} \textcolor{grey5}{\smaller (details follow)} + \vspace{1em} + \item Tight relation to \alert{linear problems \& PDEs} + \begin{itemize} + \item Convergence analysis (CG, iterative methods) + \item Quantum mechanics + \item Close relation to solving PDEs + \textcolor{grey5}{\smaller (details follow)} + \end{itemize} + \end{itemize} +\end{frame} diff --git a/01_Presentation/presentation.tex b/01_Presentation/presentation.tex new file mode 100644 index 0000000..875c4dd --- /dev/null +++ b/01_Presentation/presentation.tex @@ -0,0 +1,53 @@ +\documentclass[nostatus, +%aspectratio=1610 +]{MFHpresentation} +\usepackage{MFHcolours} +\usepackage{MFHscience} +\usepackage{MFHabbr} +\usetikzlibrary{positioning,arrows,backgrounds,fit,calc,tikzmark} +\usepackage{fontawesome} +\usepackage{multirow} +\usepackage{bbold} +\usepackage{bm} +\usepackage{marvosym} +\usepackage{ifthen} +\usepackage{transparent} + +\usetheme{MFHepfl} + +%------------------------------------------------------------------ + +\newcommand{\matmat}{\raisebox{-0.35em}{\includegraphics[height=1.45em]{img/matmat.png}}\xspace} +\newcommand{\julia}{\raisebox{-0.21em}{\includegraphics[height=1.2em]{img/julia.pdf}}\xspace} + +%------------------------------------------------------------------ + +\title{ + Error control in scientific modelling\\[-0.2em] + \textcolor{black}{\smaller[1.5] (with a focus on eigenvalue problems)} +} +\author{Michael F. Herbst, Niklas Schmitz} +\date{12 September 2024} + +\institute{Mathematics for Materials Modelling (\href{https://matmat.org}{\texttt{matmat.org}}), EPFL} + +\titlegraphic{ + \includegraphics[width=0.4\textwidth]{img/si_band_errors.pdf} + \qquad + \includegraphics[width=0.25\textwidth]{img/matmat.png} +} + +\begin{document} +\setbeamerfont{footnote}{size=\tiny} + +\showlogotrue +\begin{frame}[plain] + \titlepage + \addtocounter{framenumber}{-1} +\end{frame} + +\showlogofalse +\input{motivation} +\input{course} +\input{about_you} +\end{document} diff --git a/01_Presentation/presentation.tex.latexmain b/01_Presentation/presentation.tex.latexmain new file mode 100644 index 0000000..e69de29 diff --git a/02_Laplace_error_sources.html b/02_Laplace_error_sources.html new file mode 100644 index 0000000..ae1d185 --- /dev/null +++ b/02_Laplace_error_sources.html @@ -0,0 +1,20 @@ + + + + + + + + + + +

\ No newline at end of file diff --git a/02_Laplace_error_sources.jl b/02_Laplace_error_sources.jl new file mode 100644 index 0000000..ea84189 --- /dev/null +++ b/02_Laplace_error_sources.jl @@ -0,0 +1,1641 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═║ 325ad613-8019-4e7c-9218-ebfbe7ccbd19 +begin + using HypertextLiteral + using PlutoUI + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ d0f24091-b9d3-4c3a-8b43-935e2b9ae607 +using LinearAlgebra + +# ╔═║ 88a0697e-03b6-479c-8d49-99adda3e0e97 +using Plots + +# ╔═║ c5439ba8-c5e1-491d-88d1-1a079e0a1541 +md""" +# The Dirichlet Laplacian in an unbounded domain +""" + +# ╔═║ 49f8f904-57cd-4c41-9769-b28f01a6d048 +md""" +In the previous lectures we saw that eigenvalue problems involving the Laplace operator $-\Delta$ occur naturally both when solving problems in continuum mechanics and quantum physics. +Moreover, we saw that in quantum mechanics the ground state, i.e. the $\Psi$ associated with the smallest eigenvalue of a Hamiltonian $-\Delta + V$ are of key importance. + +To start our journey we will thus consider the problem of computing the eigenspectrum of the Laplace operator on an unbounded domain: All the way from the continuous setup to the number crunching on the computer. +Our goal is to keep rigorously track of the error contributions along the way as we introduce more and more approximations. +""" + +# ╔═║ 03a20311-7ff9-4177-9a0d-d85433a57ae0 +md""" +## Triangle inequality + +The triangle inequality is a common ingredient in numerical analysis to decompose error contributions. As you may remember from linear algebra, if you have two vectors $u$ any $v$, than the triangle inequality reads that +```math +\|u + v\| \leq \|u\| + \|v\|, +``` +where $\|\,\cdot\,\|$ denotes some norm from the vector space of $u$ and $v$. + +Now imagine we have a two-fold computational procedure. +The first, defined by the function $y = b(x)$ and the second taking +the result of this operation leading to $z = s(y)$. +To be more concrete you can think of $b$ as the method used to discretise +the problem on a finite basis and of $s$ as solving it somehow. + +Let's say both of these tasks introduce an error, +i.e. instead of $b(x)$ we actually perform $\tilde{b}(x)$ +and instead of $s(y)$ we perfom $\tilde{s}(y)$. +Overall we will thus obtain $\tilde{z}=\tilde{s}(\tilde{b}(x))$ +instead of the $z$ we are actally after. + +Usually it is much easier to separately understand the error +between $s$ and $\tilde{s}$ and between $b$ and $\tilde{b}$ +rather than directly estimating the error between $z$ and $\tilde{z}$. +The triangle inequality shows how this is feasible: +```math +\begin{aligned} +\|z - \tilde{z}\| &= \| s(b(x)) - \tilde{s}(\tilde{b}(x)) \| \\ +&= \| s(b(x)) - \tilde{s}(b(x)) + \tilde{s}(b(x)) - \tilde{s}(\tilde{b}(x)) \| \\ +&\leq \| s(b(x)) - \tilde{s}(b(x)) \| + \|\tilde{s}(b(x)) - \tilde{s}(\tilde{b}(x)) \| +\end{aligned} +``` +The first term now measures the error introduced between $s$ and $\tilde{s}$ +and the second measures the error $b$ and $\tilde{b}$. + +Let's make this more clear for the Laplacian example. +""" + +# ╔═║ 5f84e5cd-5c53-42ec-88cc-aeb9f09387d8 +TODO(md""" +Some graphics illustrating the domain (infinite versus cut, discretisation points) and the error contributions in a bar or so could be useful +""") + +# ╔═║ 9272964b-07ad-4378-8267-09ea71ac2fb6 +md""" +## Continuous setting + +To start our journey into illustrating the numerical challenges when computing eigenspectra, we consider the following problem: + +**Problem 1:** Find the smallest member of the spectrum of $- \Delta$ on the **unbounded domain** $\mathbb{R}$, i.e. the entire real line. + +In quantum mechanics this problem corresponds to the free particle Hamiltonian. From our discussion so far our ansatz would be to seek functions $\Psi$, which satisfy +```math +- \Delta \Psi = - \frac{d}{d x^2} \Psi = \lambda \Psi. +``` +As we saw earlier the eigenfunctions of a Hamiltonian are the wave function and its modulus square has the interpretation of a probability density. It thus needs to be square integrable, i.e. $\Psi \in L^2(\mathbb{R})$. + +By inspection one identifies the above equation to admit as solutions the plane waves +```math +\Psi_k(x) = e^{\pm ikx} +``` +respectively with apparent eigenvalue $\lambda = k^2$. However, clearly +```math +\int_{\mathbb{R}} |\Psi_k(x)|^2 d x = +\int_{\mathbb{R}} 1 d x +``` +which is not square-integrable. So in fact $- \Delta$ does not have **any eigenfunction**. + +However, it does possess a **spectrum**, which loosely speaking is defined as those points $z \in \mathbb{C}$ for which $-\Delta - z I$ is not bijective. +Clearly for any $\lambda \ge 0$ +```math + (-\Delta - \lambda I) \, e^{i x \sqrt{\lambda}} = 0, +``` +such that $\Psi_{\sqrt{\lambda}}$ is in the kernel of $(-\Delta - \lambda I)$, making it not even injective. + +We will discuss some consequences of such subtleties for approximating the spectra of continuous operators in more details in the second half of the lecture. + +Regarding our tasks of computing the smallest member of the spectrum of $-\Delta$, it is clear from our discussion this is just $\lambda_\ast = 0$, so an ideal test case to enter our discussion into the different error contributions. +""" + +# ╔═║ 3b0bcfad-dfa6-477a-bbe1-bbf951aada68 +md""" +## Model error + +For now to avoid any problems due to the unbounded domain we set ourselves a fixed, large $a$ and instead consider a bounded domain $\Omega = [0, a]$ with zero Dirichlet boundary conditions. In this setup $- \Delta$ has an infinite number of eigenvalues $\lambda_n$ and associated eigenfunctions $\Psi_n \in L^2([0, a])$. We use this to model *Problem 1* by the alternative + +**Problem 2:** Find the smallest eigenvalue of $- \Delta$ on the domain $[0, a]$ and with zero Dirichlet boundary conditions, i.e. solve +```math + \begin{aligned} + - \Delta \Psi_n &= \lambda_n \Psi_n && \text{in $\Omega$}\\ + \Psi_n(0) &= \Psi_n(a) = 0 + \end{aligned} +``` +for the smallest eigenpair $(\lambda_n, \Psi_n)$. + +The eigenpairs of $- \Delta$ on this domain can be easily obtained analytically as +```math +\begin{aligned} +\lambda_n &= \frac{n^2 \pi^2}{a^2}\\ +\Psi_n(x) &= \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right) +\end{aligned} +``` +for $i = 1, \ldots, \infty$, where eigenvalues are sorted in *ascending order*. + +Having changed our mathematical model for the underlying phenomenon we want to study we have thus a **model error** + +```math +e_\text{model} = \left| \lambda_1 - \lambda_\ast \right| = \frac{\pi^2}{a^2}. +``` + +Note that in this case the model error was caused by a *change of boundary conditions*. More commonly in physical settings, the model error is introduced by an approximation of the equation defining the problem itself (e.g. approximate potential in Hamiltonian $-\Delta + V$). +""" + +# ╔═║ 4c14c1fe-82e4-4ec7-942f-f37568e04910 +md""" +## Discretisation error + +For cases where analytic solution strategies are not available, *Problem 2* can still not be easily solved as it involves implicitly the search over the *infinite-dimensional* space $\Psi_n \in L^2([0, a])$, from which the eigenfunctions are taken. + +A natural next step to simplify the problem further is to consider a finite-difference discretisation of *Problem 2*: We split the domain $\Omega = [0, a]$ into $N+1$ equal intervals of width $h = \frac{a}{N+1}$. The coordinates of the respective limit points of the intervals are given by $x_j = h \cdot j$, $0 \ge j \ge N+1$. + +In this representation we can approximate the second derivative of $\Psi$ at a point $x_j$ as follows: +```math +\begin{aligned} +\left(\Delta \Psi\right)(x_j) +&= \frac{d^2}{dx^2} ( \Psi' )(x_j)\\ +&\simeq \frac{\Psi(x_j+h) - \Psi(x_j) - \Big(\Psi(x_j) - \Psi(x_j-h) \Big)}{h^2}\\ +&= \frac{\Psi(x_{j+1}) - 2 \Psi(x_j) +\Psi(x_{j-1})}{h^2}\\ +&= \left(\tilde{L} \tilde{u}\right)_j, +\end{aligned} +``` +where we collected a vector $\tilde{u} \in \mathbb{R}^{N+2}$ with elements $\tilde{u}_j = \Psi(x_j)$ +and a matrix $\tilde{L} \in \mathbb{R}^{(N+2)\times(N+2)}$ with $2/h^2$ on the diagonal and $-1/h^2$ on the super- and sub-diagonal. + +We will eventually seek eigenpairs $-\tilde{L} \tilde{u} = \mu \tilde{u}$. However, due to the boundary conditions we necessarily have $u_0 = u_{N+1} = 0$. We can thus equivalently reduce the dimensionality of our problem by two and drop the first and last columns of $\tilde{L}$ as well as the zero first and last entries of $\tilde{u}$. + +This leads to the discretised problem: + +**Problem 3:** Find the smallest eigenvalue $\mu_n \in \mathbb{R}$ +and corresponding eigenfunction $u_n \in \mathbb{R}^{N}$ +of the eigenproblem +```math + H u_n = \mu_n u_n +``` +with +```math +\begin{align} + H = -L, & & \text{where}\; + L = \frac1{h^2} \left( \begin{array}{ccccc} + -2 & 1 \\ + 1 & -2 & 1 \\ + & \ddots & \ddots & \ddots \\ + && 1 & -2 & 1 \\ + &&& 1 & -2 + \end{array}\right) +\end{align} +``` + +Programmatically the matrix $L$ (the discretised Laplacian) can be obtained by the following function: +""" + +# ╔═║ e9b97d04-324e-4059-8a3c-d79829d2e4f8 +function fd_laplacian(N, a; T=Float64) + h = a / T(N+1) + diagonal = -2ones(T, N) ./ h^2 + side_diagonal = ones(T, N-1) ./ h^2 + SymTridiagonal(diagonal, side_diagonal) +end + +# ╔═║ e0031dfb-1cec-446b-a577-6c6c15f76b2d +fd_laplacian(10, 11) + +# ╔═║ f2e33ed8-d7ae-41f9-bf94-e080477cd5de +md""" +Since *Problem 3* has a comparatively simple structure, analytic solutions can also be found analytically in this case, namely +```math +\begin{aligned} + \mu_n &= \frac{4}{h^2} \sin^2\left( \frac{n\, \pi}{ 2(N+1) } \right)\\ + (u_n)_i &= \sqrt{\frac{2}{N+1}} \sin\left( \frac{n\, i\, \pi}{N+1} \right) +\end{aligned} +``` +for $n = 1, \ldots, N$, where eigenvalues are again given in *ascending order*. The lowest eigenvalue is thus +```math +\mu_1 = \frac{4 (N+1)^2 }{a^2} \sin^2\left( \frac{\pi}{ 2(N+1) } \right). +``` +Programmatically this can be obtained as: +""" + +# ╔═║ c0ab4a2b-6ffb-41d5-8318-cddc534b1154 +begin + function fd_eigenvalue(N, a, n) + h = a / (N+1) + sqrt_mu = 2 / h * sin(n * Ο€ / 2(N+1)) + sqrt_mu^2 + end + + function fd_eigenfunction(N, a, n) + h = a / (N+1) + i = 1:N + @. sqrt(2 / (N+1)) * sin(i * n * Ο€ / (N+1)) + end +end + +# ╔═║ f94d4725-51e7-4323-942c-ad8125f0b6fb +md""" +Note that we are using `BigFloat`, Julia's built-in arbitrary precision floating-point data type, to ensure the returned values are free of any rounding error. This is needed for the next section. +""" + +# ╔═║ dec7f34f-79c5-492e-b9cd-67bf7005d6c6 +let # Plot eigenfunction for one example + N = 100 + a = 1.0 + xs = (1:N) .* a ./ (N+1) + p = plot() + for n in 1:3 + Ξ»n = round(fd_eigenvalue(N, a, n); digits=2) + plot!(p, xs, fd_eigenfunction(N, a, n), label="u_$n with Ξ»_$n = $Ξ»n") + end + p +end + +# ╔═║ 38a0c072-fd9e-4786-b946-4a851f655e94 +md""" +Compared to *Problem 2* we thus have introduced a **discretisation error** of +```math +e_\text{discretisation} = \left| \lambda_1 - \mu_1 \right| = \left| \frac{\pi^2}{a^2} - \frac{4 (N+1)^2 }{a^2} \sin^2\left( \frac{\pi}{ 2(N+1) }\right) \right|. +``` +In can be easily verified analytically that this converges to zero as $N\to\infty$ in agreement with this plot: +""" + +# ╔═║ a2fdeb6f-277a-409c-8cc6-7cc2ce47a74b +let + a = 1.0 + Ns = 100:100:5000 + + Ξ»1 = Ο€^2 / a^2 + ΞΌ1 = [fd_eigenvalue(N, a, 1) for N in Ns] + + p = plot(; yaxis=:log) + plot!(p, Ns, abs.(ΞΌ1 .- Ξ»1), label="discretisation error") + hline!(p, [Ξ»1], label="model error a=$a") + plot!(p, Ns, abs.(ΞΌ1), label="discr+model error a=$a") +end + +# ╔═║ 301cd60e-e0a9-4139-af36-37b91540a2dd +md""" +Overall error is however clearly dominated by the model error. +""" + +# ╔═║ 1637720d-7182-4a9c-817a-15a880eaabfc +md""" +## Algorithm error and arithmetic error + +In order to obtain the lowest eigenvalue $\mu_1$ numerically, we can employ the inverse power method, as discussed in the exercises last week. + +We start by an implementation of the inverse power method: +""" + +# ╔═║ 22ba6c9f-8c3b-487e-bcf6-0d332aae6166 +function inverse_power_method(A; u=randn(eltype(A), size(A, 2)), + tol=1e-6, maxiter=500) + norm_Ξ”u = NaN + Afac = factorize(A) # Factorise A to make A \ x more economical + for i in 1:maxiter + u_prev = u + u = Afac \ u + normalize!(u) + norm_Ξ”u = min(norm(u - u_prev), norm(-u - u_prev)) + norm_Ξ”u < tol && break + end + ΞΌ = dot(u, A, u) + norm_Ξ”u β‰₯ tol && @warn "Inverse power not converged $norm_Ξ”x" + (; ΞΌ, u) +end + +# ╔═║ 0277750b-fa72-4cbd-b616-3f0d07213102 +md"... which we now use to compute the smallest eigenvalue of $H$:" + +# ╔═║ 1e2b9f41-87be-46ca-b34d-d2c148f52f8a +function numerical_eigenvalue(N, a; T=Float64, kwargs...) + H = -fd_laplacian(N, a; T) + inverse_power_method(H; kwargs...).ΞΌ +end + +# ╔═║ c3ab23f6-a98b-439e-8de4-ac9f8a5f4ee3 +md"Since we have the analytical solution, we can easily check the accuracy of this procedure:" + +# ╔═║ 5f232600-2871-4b5d-bd3a-f88936534b61 +let + Ns = 100:100:1000 + a = 1. + + ΞΌ_numeric = [numerical_eigenvalue(N, a) for N in Ns] + ΞΌ = [fd_eigenvalue(N, a, 1) for N in Ns] + + plot(Ns, abs.(ΞΌ_numeric .- ΞΌ); label="error", yaxis=:log) +end + +# ╔═║ 1500555d-eebf-414c-9923-ac2db8564d56 +md""" +While not very large, the numerical computation of the eigenvalue of the discretised problem introduces an additional error. There are many possible sources for this error, including: + +- **Algorithm error:** Numerical computations often involve iterative procedures, which are terminated as soon as a convergence criterion is satisfied. In our example we stop the inverse power method once the norm in the change of the eigenvector is below $10^{-6}$. This "early stopping" (compared to running ad infinitum) introduces additional errors. The errors stemming from such algorithmic choices for the numerical solution procedure we call jointly "algorithm error" +- **Arithmetic error:** The error introduced by performing computation in finite-precision floating point arithmetic. + +Albeit more common than one perhaps thinks, we will not discuss there error contributions in the lecture: +- **Hardware error:** Errors due to hardware malfunctioning or hardware bugs. +- **Programming error:** Errors due to bugs in the software implementation. + +*Note:* The terminology of the error contributions varies a little between the literature. +""" + +# ╔═║ 38f6171b-6572-4909-91b2-a3c1c85e6490 +md""" +Ignoring the latter two contributions the error shown in the plot above, i.e. the discrepancy between `fd_eigenvalue` and `numerical_eigenvalue` is the sum of the algorithm error and the arithmetic error. To estimate the two separate contributions, we will make the assumption that the `BigFloat` number format is free of arithmetic error. Note that this is not fully correct (by default uses 256bit of precision) and we will discuss better techniques later in the lecture. + +Making this assumption we estimate the algorithm error $e_\text{algorithm}$ computationally as: +""" + +# ╔═║ e00f4a80-7d09-4b04-87c1-fafd5c080d48 +begin + N = 1000 + a = 1. + + u = randn(N) + ΞΌ1_fp64 = numerical_eigenvalue(N, a; T=Float64, u) # ΞΌ_1^{(fp64)} + ΞΌ1_big = numerical_eigenvalue(N, a; T=BigFloat, u=BigFloat.(u)) # ΞΌ_1^{(big)} + ΞΌ1_exact = fd_eigenvalue(N, a, 1) + + e_algorithm = abs(ΞΌ1_big - ΞΌ1_exact) +end + +# ╔═║ eb9b957c-3403-4ee3-99cc-b7991e82bac3 +md"... and we obtain the remaining error as the arithmetic error $e_\text{arithmetic}$:" + +# ╔═║ 91f11ab7-c2a3-40f8-862a-d90f15652c5a +e_arithmetic = abs(ΞΌ1_big - ΞΌ1_fp64) + +# ╔═║ b5d7df6e-6858-444e-9b8d-533264ae0465 +md"""... and the total error for N=1000 and a=1 as:""" + +# ╔═║ 0c7cc94a-e464-4dc8-869e-909befc9ab90 +e_total = abs(ΞΌ1_fp64) + +# ╔═║ 2cd6aaec-a049-47e0-87c1-5c18cd6e0c42 +md""" +## Separating error contributions + +If we denote by $\mu_1^{(\text{fp64})}$ the estimate for the lowest eigenvalue of $-\Delta$ on an unbounded domain returned by `numerical_eigenvalue` using double precision and by $\mu_1^{(\text{big})}$, the total error compared to the exact value $\lambda_\ast$ is simply $|\lambda_\ast - \mu_1^{(\text{fp64})}|$. + +Since in this case $\lambda_\ast$ is known, we are in the fortunate situation that we can directly compute the error. However, for more challenging problems this is not feasible. One goal of the lecture will be to identify techniques, which allow estimating an *upper bound* for this error. A crucial strategy for this purpose is to separate the total error into the contributions discussed above. As we will see for each contribution separately it will be much easier to devise analytical or computational techniques to estimate their respective errors. + +The key strategy is to employ the triangle inequality. Following the notation from above: + +```math +\begin{aligned} + |\lambda_\ast - \mu_1^{\text{(fp64)}}| +&≀ |\lambda_\ast - \lambda_1| + |\lambda_1 - \mu_1| \hspace{2.1ex} + |\mu_1 - \mu_1^{\text{(big)}}| + |\mu_1^{\text{(big)}} - \mu_1^{\text{(fp64)}}|\\ +&= e_\text{model} \hspace{3.4ex} + e_\text{discretisation} + e_\text{algorithm} \hspace{3ex} + e_\text{arithmetic} +\end{aligned} +``` +""" + +# ╔═║ 482582b0-1ea2-43b5-b97d-2760210eb6a4 +TableOfContents() + +# ╔═║ c2d88d22-e423-4942-87f3-812c35e693ac +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 330) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoTeachingTools = 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literal 0 HcmV?d00001 diff --git a/03_Matrix_eigenproblems.html b/03_Matrix_eigenproblems.html new file mode 100644 index 0000000..fa0c382 --- /dev/null +++ b/03_Matrix_eigenproblems.html @@ -0,0 +1,20 @@ + + + + + + + + + + +

\ No newline at end of file diff --git a/03_Matrix_eigenproblems.jl b/03_Matrix_eigenproblems.jl new file mode 100644 index 0000000..cadec3a --- /dev/null +++ b/03_Matrix_eigenproblems.jl @@ -0,0 +1,975 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═║ 317ff75e-d718-11ee-1bbb-1374836729ec +begin + using PlutoUI + using HypertextLiteral + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ cec2f53f-c4a3-4bb8-bc67-99243a3e9d4f +md""" +# Matrix Eigenproblems +""" + +# ╔═║ 9bf047e1-624e-4656-9c4d-d1021617ee9d +md""" +## Preliminaries +(We follow Saad 2011) + +!!! note "Definitions (Eigenvalue/vector/space, spectrum, spectral radius)" + Let $A \in \mathbb C^{n \times n}$ be a square matrix. + + - $Ξ» \in \mathbb C$ is an *eigenvalue* of $A$ if and only if $A - \lambda I$ is *not* invertible, i.e. if $\ker(A - \lambda I) \neq \{0\}$. + + - A non-zero element $v \in \ker (A- \lambda I)$, i.e. $A v = \lambda v$, is called an *eigenvector*. + + - The space $\eigenspace_A(\lambda) = \ker (A - \lambda I)$ is called *eigenspace*. + + - The set of all eigenvalues $\sigma(A)$ is called the *spectrum*. + + - We can further define the *spectral radius* $\spectralradius(A)$ as $\max_{\lambda \in \sigma(A)} | \lambda |$. + +""" + +# ╔═║ 0d33d753-d27b-43a5-aef2-6a020fb7be5b +md""" +!!! note "Definitions (Trace, kernel, image)" + Let $A \in \mathbb C^{n \times n}$. We further define the following quantities + ```math + \begin{align} + &\text{Trace} & \text{tr} (A) &\coloneqq \sum_i A_{ii} + \\ + &\text{Kernel} & \ker (A) &\coloneqq \{ v \in \mathbb C^n \mid Av = 0\} + \\ + &\text{Image} & \im (A) &\coloneqq \{ w \in \mathbb C^n \mid \exists v \in \mathbb C^n \text{ s.t. } Av = w \} + \end{align} + ``` + +!!! tip "Remark" + $\ker(A)$ and $\im (A)$ are subspaces of $\mathbb C^n$, and + ```math + \dim ( \ker(A)) + \dim (\im(A)) = n. + ``` +""" + +# ╔═║ cac1fb8f-f495-4ddb-ace0-58c17e39ebc3 +md""" +!!! note "Theorem 1" + Let $A \in \mathbb C^{n \times n}$. $A$ is injective if and only if $\ker (A) = \{ 0 \}$. + +> *Proof.* +> +> $\boxed \Rightarrow$ : Let $v \in \ker (A)$. Then +> ```math +> \begin{align} +> Av = 0 &\Rightarrow A(-v) = - Av = 0 +> \\ &\hspace{-0.8em} \stackrel{\text{injective}}{\Rightarrow} v = -v +> \\ &\Rightarrow v = 0 +> \end{align} +> ``` +> +> $\boxed \Leftarrow$ : Let $A v_1 = A v_2$. Then +> ```math +> \begin{align} +> A(v_1 - v_2) = A v_1 - A v_2 = 0 &\Rightarrow v_1 - v_2 \in \ker (A) +> \\&\hspace{-1.4em} \stackrel{\ker (A) = \{ 0\}}{\Rightarrow} v_1 - v_2 = 0 +> \\ &\Rightarrow v_1 = v_2 +> \\ & && & \square +> \end{align} +> ``` + +""" + +# ╔═║ b1507c1f-1f1d-41f9-8b89-5d59a9a90e75 +md""" +## Matrix norms + +Clearly, $A \in \mathbb C^{n \times n}$ is a vector space. +What are good norms for it ? + + +""" + +# ╔═║ 88e4ba37-7420-4170-b9bc-3229cb257335 +md""" + +- Recall the conditions a norm should satisfy, for all $A \in \mathbb C^{n\times n}$ : + ```math + \begin{align} + \| A \| &\geq 0 && \| A \| = 0 \iff A = 0 + \\ + \| \alpha A \| &= | \alpha| \| A \| && \forall \alpha \in \mathbb C + \\ + \| A + B \| &\leq \| A \| + \| B \| && \forall B \in \mathbb C^{n \times n} \tag{$\triangle$} + \end{align} + ``` + The third condition is known as the triangle inequality and sometimes denoted as $\triangle$. + + +- The $p$-norms of $\mathbb C^n$ ($\| v \|_p = (\sum_{i=1}^n |v_i|^p)^{1/p} \quad \forall v \in \mathbb C^n$) inherit a matrix norm for $p \geq 1$ or $p = \infty$ as + ```math + \| A \|_p = \max_{0 \neq v \in \mathbb C^n} \frac{\| A v \|_p}{\|v\|_p}, + ``` + the so-called *induced matrix norm*. + +- They are always consistent with the associated vector norm, i.e. + ```math + \| A v \|_p = \| v \|_p \frac{\| A v \|_p }{\|v\|_p} \leq \| v\|_p \max_{0 \neq v \in \mathbb C^n} \frac{\| A v \|_p}{\| v \|_p} = \| A \|_p \| v \|_p. + ``` + Further, we obtain + ```math + \| A B \|_p = \max_{x \neq v \in \mathbb C^n} \frac{\| A B v \|_p}{\| v \|_p} \leq \| A \|_p \max_{0 \neq v \in \mathbb C^n } \frac{\| B v \|_p}{\| v\|_p} = \| A \|_p \| B \|_p + ``` + Thus, + ```math + \begin{align} + \| A ^2 \|_p &\leq \| A \|_p^2. + \end{align} + ``` +- More generally, + ```math + \begin{align} + \| A ^k \|_p &\leq \| A \|_p^k. + \end{align} + ``` +""" + +# ╔═║ e10a7029-411a-4c06-abd7-dc5f835fe7db +md""" +Here are some explicit definitions for induced matrix norms +```math +\begin{align} +\| A \|_1 &= \max_{j = 1, \dots, n} \sum_{i=1}^n | A_{ij} | \tag{Column-sum norm} +\\ + \| A \|_\infty &= \max_{i = 1, \dots, n} \sum_{j=1}^n |A_{ij}| \tag{Row-sum norm} +\\ +\| A\|_2 &= \sqrt{\spectralradius (A A^H)} = \sqrt{\spectralradius(A^H A)} +\end{align} +``` +Here, $A^H$ denotes the Hermitian conjugate, i.e. $A^H = \overline{ A ^T} = (\bar A ) ^T$. +For those more familiar with linear algebra over the reals, Hermitian matrices ($A^H = A$) generalize many properties of real symmetric matrices, which are themselves all Hermitian. +!!! tip "Remark" + $\| A \|_2$ is also equal to the largest singular value of $A$. + In addition, since $A^HA$ (and $A A ^H$) is a Hermitian positive semi-definite matrix, the spectral radius of $A ^H A$ (and $A A ^H$) is simply the largest eigenvalue of $A^HA$, and we can write + ```math + \| A \|_2 = \sqrt{\lambda_{max} ( A ^H A )} = \sqrt{\lambda_{max} ( A A ^H)} + ``` +""" + +# ╔═║ 9e95ff3b-ca5e-4a88-a466-bf59b0578bd1 +md""" +Another important matrix norm is the *Frobenius norm* +```math +\| A \|_F = \sqrt{\mathrm{tr} ( A A ^H)} = \sqrt{\sum_{i,j = 1}^n | A _{ij}^2|} +``` +Its relevance will become clear once we consider the equivalence of matrix norms, which we treat in the next section. +""" + +# ╔═║ 6aef964d-650c-460c-807d-9afcf6d7475e +md""" +## Equivalence of norms + +!!! note "Definition (Norm equivalence)" + The norms $\| \cdot \|_p$ and $\| \cdot \|_q$ over a space $V$ are equivalent if there exists positive constants $\alpha, \beta > 0$ such that for all $A \in V$ + ```math + \alpha \| A \|_p \leq \| A \|_q \leq \beta \| A \|_p + ``` + +!!! note "Proposition 2 (Equivalence of Frobenius norm and 2-norm)" + We consider the vector space of matrices $A \in \mathbb C^{n \times n}$. + The Frobenius norm $\| \cdot \|_F$ and the spectral norm $\| \cdot \|_2$ are equivalent. + Moreover, + ```math + \| A \|_2 \stackrel{(a)}{\leq} \| A \|_F \stackrel{(b)}{\leq} \sqrt{n} \| A \|_2 + ``` + +This relationship is relevant as $\| A \|_2$ proves useful in theory but is hard to compute, unlike $\| A \|_F$ for which computation is quite straightforward. + +> *Proof.* +> (a) is left as an exercise. For (b), we have +> ```math +> \begin{align} +> \| A \|_F^2 = \text{tr} ( A ^H A ) = \sum_{i=1}^n \lambda_i( A ^H A) \leq \sum_{i=1}^n \lambda_{max} ( A ^H A ) = n \| A \|_2^2 +> && \square +> \end{align} +> ``` +""" + +# ╔═║ bc5beb41-903f-439f-ac2f-40b523945563 +md""" +## Similarity + +!!! note "Definition (Similarity and diagonalisability)" + $A \in \mathbb C^{n \times n}$ is *similar* to $B \in \mathbb C^{n \times n}$ if there exists an invertible $S \in \mathbb C^{n \times n}$ such that $S^{-1} A S = B$. + + If $S$ is *unitary*, i.e. $S^{-1} = S^H$, then $A$ and $B$ are *unitarily similar*. + + $A \in \mathbb C^{n \times n}$ is *diagonalisable* if it is similar to a diagonal matrix. + +!!! tip "Remark (Eigenvalues of similar matrices)" + Similar matrices have equal eigenvalues of equal multiplicities. + If $(\lambda, v)$ is an eigenpair of $A$ and $B = S^{-1} A S$, then $(\lambda, S^{-1} v)$ is an eigenpair of $B$. + Indeed, since + ```math + B S^{-1} v = S^{-1} A S S^{-1} v = S^{-1} A v = \lambda S^{-1} v. + ``` + Thus, the eigenvalues of $A$ and $B$ are equal and their multiplicities the same. + +!!! tip "Remark" + If $A$ and $B$ are unitarily similar and $A$ is Hermitian, so is $B$, since + ```math + B = U^{-1} A U = U^H A U. + ``` + +!!! tip "Remark" + $A \in \mathbb C^{n \times n}$ is diagonalizable if and only is there exists $n$ linearly independent eigenvectors $v_i, i = 1, \dots n$. + To see this, we collect all eigenvectors as columns + ```math + S = \begin{bmatrix} v_1 & v_2 & \dots & v_n \\ \downarrow & \downarrow & & \downarrow \end{bmatrix} + ``` + implying $AS = S \Lambda$ with $\Lambda = \text{diag} (\lambda_1, \dots, \lambda_n)$. + Since $v_i$ are linearly independent, $S$ is invertible, thus $S^{-1} A S = \Lambda$. + +!!! tip "Remark (Invariance of trace)" + The trace is invariant under a similarity transform. + Indeed, noticing $\text{tr} (ABC) = \text{tr} (BCA)$, we obtain + ```math + \text{tr} ( S ^{-1} A S ) = \text{tr} (A S S^{-1}) = \text{tr} (A) + ``` +""" + +# ╔═║ 4b0bef6c-39da-458a-bd72-5db98b972435 +md""" +## Quadratic forms and the spectral theorem + +""" + +# ╔═║ 8e87aa27-131a-4fc2-b837-fce855fdc8d4 +md""" +!!! note "Definition (Quadratic form)" + The polynomial $q_A : \mathbb C^n \to \mathbb C$ + ```math + q_A(x) = \sum_{i,j=1}^n \overline{x_i} A_{ij} x_j = \langle x , A x \rangle + ``` + associated to a matrix $A$ is called *quadratic form* of $A$. +""" + +# ╔═║ 82281d67-5824-4974-a4ae-504f5f91c338 +md""" +Based on the quadratic form we can show + +!!! note "Lemma 3" + Every matrix $A \in \mathbb C^{n \times n}$ has at least one eigenvalue $\lambda \in \mathbb C$. + +> *Proof.* +> - Consider +> ```math +> \min_{x \in \mathbb C^n,\, g(x) = 0} q_A(x) \quad \text{with}\quad g(x) = \langle x,x \rangle -1. +> ``` +> +> - $q_A$ is a polynomial, and thus continuous. +> - Let $S = \{ x \in \mathbb C^n \mid g(x) = 0 \} \neq \varnothing$, which is closed and bounded. +> - Therefore, $q_A$ must have a minimum over $S$ at $v \in S$ by the extreme value theorem (proved in the appendix). +> - Further, $q_A$ and $g$ are complex (Wirtinger) differentiable. +> - Thus, by the Lagrange multiplier theorem there exists $\lambda \in \mathbb C$ such that +> ```math +> \begin{align} +> &\nabla q_A(v) = \lambda \nabla g(v) +> \\ +> &\begin{cases} +> A^T \bar v = \lambda \bar v & \qquad \text{(Wirtinger derivative wrt }x) \\ A v = \lambda v & \qquad \text{(Wirtinger derivative wrt } \bar x) +> \end{cases} +> \end{align} +> ``` +> - Hence, $\lambda$ is an eigenvalue of A. $\hspace{9.0cm} \square$ +""" + +# ╔═║ abae949c-3d70-4d16-ac3b-18397f12fa51 +md""" +Our special interest are Hermitian matrices, particularly due to the importance of self-adjoint operators in quantum mechanics and other physically relevant problems. +As a reminder, Hermitian matrices satisfy $A^H = A$, i.e. $\langle u, Au \rangle = \langle Au, u \rangle$. +For these, we have two key results : + +!!! note "Theorem 4" + Eigenvalues of Hermitian matrices are real. + +> *Proof.* +> If $(\lambda, u)$ is an eigenpair of Hermitian $A \in \mathbb C^{n \times n}$ with $\langle u, u \rangle = 1$, then +> ```math +> \begin{align} +> \lambda &= \lambda \langle u, u \rangle +> \\ &= \langle u, A u \rangle +> \\ &= \langle A u, u \rangle +> \\ &= \overline{\langle u, A u \rangle} = \bar \lambda &&& \square +> \end{align} +> ``` +""" + +# ╔═║ b7ad00f2-4112-458c-bf79-6c78a722fde2 +md""" +!!! note "Lemma 5" + Let $A \in \mathbb C^{n \times n}$ be Hermitian and $W$ an invariant subspace of $A$, i.e. $\forall x \in W, A x \in W.$ + Then, $W^\perp = \{ x \mid \langle x,y \rangle = 0 \quad \forall y \in W\}$ is also an invariant subspace, i.e. if $x \in W^\perp$, then $A x \in W^\perp$. + +> *Proof.* +> Consider $x \in W^\perp$. For all $y \in W$, we have +> $\langle y , A x \rangle = \langle A y , x \rangle = 0$ since $A y \in W$. +> Thus, $A x \in W^\perp$. + +""" + +# ╔═║ 7b4df739-334e-4c30-b42e-38395848a927 +md""" +!!! note "Theorem 6 (Spectral theorem)" + If $A \in \mathbb C^{n \times n}$ is a Hermitian matrix, then there exists a decomposition $A = U \Lambda U^H$ into a real diagonal matrix $\Lambda = \text{diag} (\lambda_1, \dots, \lambda_n)$ and a unitary matrix $U = (u_1, \dots, u_n)$ with $A u_i = \lambda_i u_i \quad \forall i = 1, \dots, n.$ + +> *Proof* by induction over $n$. +> +> $n = 1$ : Trivial, just take $\Lambda_{11} = A_{11}$ (the only element of $A$), $u_{11} = 1$. +> +> $n \geq 1$ : +> - By Lemma 3 and Theorem 4, there exists $\lambda_1 \in \mathbb R$ and a $v_1 \in \mathbb C^n$ such that $A v_1 = \lambda_1 v_1$. +> - Let $W = \text{span} \{v_1\}$ and $\{ v_2 ,\dots, v_n \}$ be an orthonormal basis for $W^\perp$, and collect $V = (v_1, \dots, v_n)$. +> - Then, +> ```math +> V^H A V = \begin{pmatrix} \lambda_1 & C \\ 0 & B \end{pmatrix} +> ``` +> where $B \in \mathbb C^{(n-1) \times (n-1)}, C \in \mathbb C^{1 \times (n-1)}$. +> - As $W$ is an invariant subspace of $A$, so is $W^\perp$ by Lemma 5, and thus $C = 0$. +> - By induction hypothesis $B = Q D Q^T$ with $D$ real and diagonal and $Q$ unitary. +> The decomposition is thus obtained as +> ```math +> \begin{align} +> U &= V \begin{pmatrix} 1 & 0 \\ 0 & Q \end{pmatrix} +> &\qquad +> \Lambda &= \begin{pmatrix} \lambda & 0 \\ 0 & D \end{pmatrix} && && \square +> \end{align} +> ``` +""" + +# ╔═║ 7f03c100-5a14-4a66-b28b-3de2907775d1 +md""" +!!! tip "Remark (Normal matrices)" + The spectral theorem generalizes to normal matrices (with $A A^H = A^H A$), where the key point is that in this case the spectral theorem can be applied to the Hermitian matrix $A A^H$. + +!!! tip "Remark (Eigenvectors of real symmetric matrices)" + The eigenvectors of *real* symmetric matrices may be chosen to be real as well. + +""" + +# ╔═║ f03007bf-14b7-4d2f-98cd-cfd17e3a97c8 +md""" +## Rayleigh quotient and the min-max principle. + +For the existence of eigenvalues in matrix $A$, the quadratic form $q _A(x) = \langle x , Ax \rangle$ played a crucial role. +Essentially its extrema for $\| x \| = 1$ gave us the existence of eigenvalues (proof of Lemma 3). + +Another important and related object is the *Rayleigh quotient* +```math +R_A(x) = \frac{q_A(x)}{\| x \|^2} = \frac{\langle x , Ax \rangle}{\langle x,x \rangle}. +``` +In this section, we will characterize its extrema. + +!!! tip "Remark" + $R_A(x)$ is bounded in $\mathbb C^n$ since $| R_A(x) | \leq \| A \|_2$. + +""" + +# ╔═║ a1caae5f-5ecd-4177-bd39-2c606d068629 +md""" +We focus on Hermitian matrices $A \in \mathbb C^{n \times n}.$ + +- For these, $q_A(x)$ is always real, since $\langle x, Ax \rangle = \overline{\langle A x,x \rangle} = \overline{\langle x , Ax \rangle}$. + Therefore, $R_A(x)$ is real as well. +- The converse is also true and is the subject of an exercise. + +- We also notice that the Rayleigh quotient of an eigenvector $v_k$ gives the corresponding eigenvalue $\lambda_k$. + Indeed, + ```math + \frac{\langle v_k, A v_k \rangle}{\langle v_k, v_k \rangle} = \frac{\langle v_k, \lambda_k v_k \rangle}{\langle v_k, v_k \rangle} = \lambda_k \frac{\langle v_k, v_k \rangle}{\langle v_k, v_k \rangle} = \lambda_k. + ``` + +- As the eigenvalues of $A$ are real, we can order them ascendingly as + ```math + \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_n. + ``` + This is the convention we will usually employ. + Note that in this convention we enumerate all $n$ eigenvalues, even if they have the same value, i.e. $\lambda_i = \lambda_{i+1}$. + The number of repeated eigenvalues gives their multiplicity. + +- This convention admits the following important characterization : +""" + +# ╔═║ 109fb7ba-a0fd-4926-9b42-151cd98bf731 +md""" +!!! note "Theorem 7 (Courant-Fisher min-max principle)" + The $k$-th eigenvalue (in ascending order, counting multiplicities) of a Hermitian matrix $A \in \mathbb C^{n \times n}$ is characterized by + ```math + \begin{align} + \lambda_k &= \max_{\substack{S \subset \mathbb C^n \\ \dim (S) = n - k + 1}} \min_{0 \neq v \in S} R_A(v) + \\ + \lambda_k &= \min_{\substack{S \subset \mathbb C^n \\ \dim (S) = k}} \max_{0 \neq v \in S} R_A(v) + \end{align} + ``` + where $S$ is an $n -k +1$-dimensional or a $k$-dimensional subspace of $\mathbb C^n$, respectively. +""" + +# ╔═║ cf4669a6-5347-4c3b-b284-28254259032e +md""" +> *Proof.* +> We start by proving the first statement. +> Since $A$ is Hermitian it is unitarily similar to a diagonal matrix of its eigenvalues +> ```math +> \Lambda = \textrm{diag} (\lambda_1, \dots, \lambda_n) = U^{-1} A U +> ``` +> with $U$ containing the eigenvectors as columns. +> +> Let $S_k$ denote the span of the first $k$ eigenvectors in $U$, i.e. $S_k = \textrm{span} (v_1, \dots, v_k)$. +> Since the eigenvectors are linearly independent, $\dim S_k = k$. +> +> We consider an *arbitrary* $(n-k+1)$-dimensional subspace $S$. +> Since $\dim (S) = n-k+1$ and $\dim S_k = k$ thus the sum exceeding $n$, +> their intersection cannot reduce to $\{ 0 \}$, i.e. $S \cap S_k \neq \{ 0 \}$. +> Therefore, there exists $x \in S \cap S_k$ with $x \neq 0$. +> We write +> ```math +> x = \sum_{i=1}^k c_i v_i +> ``` +> such that +> ```math +> R_A(x) = \frac{\langle x, Ax \rangle}{\langle x,x \rangle} = \frac{\sum_{i=1}^k |c_i|^2 \lambda_i}{\sum_{i=1}^k |c_i|^2} \leq \lambda_k \frac{\sum_{i=1}^k |c_i|^2}{\sum_{i=1}^k |c_i|^2} = \lambda_k +> ``` +> since by construction $\lambda_i \leq \lambda_k$ for $i = 1, \dots, k$. +> Thus, denoting the minimum of $R_A$ over subspace $S$ as $\mu(S)$, we obtain +> ```math +> \tag{a} +> \mu(S) \coloneqq \min_{0 \neq v \in S} R_A(v) \leq R_A(x) \leq \lambda_k. +> ``` +> +> On the other hand, now consider the *particular* $(n-k+1)$-dimensional subspace $\tilde S = \textrm{span} (v_k, \dots, v_n)$. +> Similarly to our previous construction, for any $\tilde x = \sum_{i=k}^n \tilde c_i v_i \in \tilde S$ it holds +> ```math +> R_A(\tilde x) = \frac{\sum_{i=k}^n \tilde |c_i|^2 \lambda_i}{\sum_{i=k}^n \tilde |c_i|^2} \geq \lambda_k +> ``` +> and thus +> ```math +> \tag{b} +> \mu(\tilde S) = \min_{0 \neq \tilde x \in \tilde S} R_A(\tilde x) \geq \lambda_k. +> ``` +> +> Therefore according to (a) *any* $(n-k+1)$-dimensional subspace $S$ has $\mu(S) \leq \lambda_k$, while according to (b) the special subspace $\tilde{S}$ of this size even has $\mu(\tilde{S}) \geq \lambda_k$, such that +> ```math +> \max_{\substack{S \subset \mathbb C^n \\ \dim (S) = n - k + 1}} \mu(S) = \lambda_k, +> ``` +> which proves the first result. +> The second can be proven using an analogous strategy. $\qquad \square$ +""" + +# ╔═║ a79a3ad9-f7ec-43f6-ae77-70d18d3bbc40 +md""" +!!! tip "Remark (λ₁ and Ξ»β‚™)" + As a direct consequence we obtain + ```math + \begin{align} + \lambda_1 &= \max_{\substack{S \subset \mathbb C^n \\ \dim (S) = n}} \min_{0 \neq v \in S} R_A(v) = \min_{0 \neq v \in \mathbb C^n} R_A(v) + \\ + \lambda_n &= \min_{\substack{S \subset \mathbb C^n \\ \dim (S) = n}} \max_{0 \neq v \in S} R_A(v) = \max_{0 \neq v \in S} R_A(v) + \end{align} + ``` +""" + +# ╔═║ 2dbc7c07-3c25-4968-be48-4bd8a2b0fb8b +md""" +A further characterization is obtained as a corollary + +!!! note "Corollary 8" + The $k$-th eigenvalue (in ascending order, counting multiplicities) of a Hermitian matrix $A \in \mathbb C^{n \times n}$ is given by + ```math + \lambda_k = \frac{\langle v_k, A v_k \rangle}{\langle v_k, v_k \rangle} = \min_{\substack{0 \neq x \in \mathbb C^n \\ x \perp S_{k-1}}} R_A(x) + ``` + where $S_{k-1} = \textrm{span} (v_1, \dots, v_{k-1})$. + The $k$-th eigenvector can be further obtained as the maximizer of $R_A$, orthogonal to all following eigenvectors (with larger eigenvalues). +""" + +# ╔═║ 36589e2a-2a7d-46f0-ba0b-1a6285940941 +md""" +Overall, Theorem 7 and the Rayleigh quotient provide the theoretical framework to compute appropriate eigenpairs by minimization in subspaces. +We will explain this algorithmically in the next lecture. + +The study of the Rayleigh quotient also provides direct insight into matrix properties, e.g. for a Hermitian matrix to be positive definite it is necessary and sufficient that $R_A(x) > 0$ for all $x \in \mathbb C^n$ with $x \neq 0$. +Similarly, to be positive semidefinite we need $R_A(x) \geq 0$. +""" + +# ╔═║ f9ed8b9f-1be9-465f-98e6-9d1bdd0627b1 +md""" +## Appendix : Proof of the extreme value theorem +""" + +# ╔═║ 38c3ca34-7837-4c14-8ea4-a8a5a3daa415 +md""" +We start by proving the boundedness theorem. + +!!! note "Theorem 9 (Boundedness theorem)" + If $f : [a,b] \to \mathbb R$ is continuous on $[a,b]$, then it is bounded on $[a,b]$. + +> *Proof.* +> - Suppose $f$ is not bounded by above on $[a,b]$. +> - Then, for every $n \in \mathbb N$ there exists $x_n \in [a,b]$ with $f(x_n) > n$. +> This defines a sequence $(x_n)$. +> - As $[a,b]$ is bounded, there must be a convening subsequence $(x_{n_k})$ of $(x_n)$ with limit $x$ by the Bolzano-Weirestrass theorem. +> - As $[a,b]$ is closed, we must have $x \in [a,b]$. +> - Further, $f$ is continuous, thus $f(x_{n_k}) \to f(x)$ as $x_{n_k} \to x$. +> - But, $f(x_{n_k}) > n \geq k$, therefore $f(x_{n_k}) \to \infty$ as $x_{n_k} \to x$. +> - This contradicts our initial statement. +> +> We can consider an analogous argument for the lower bound. $\hspace{4cm} \square$ +""" + +# ╔═║ 7aaa09c1-db9f-4791-807e-50cc19ffaf8a +md""" +!!! note "Theorem 10 (Extreme value theorem)" + A continuous real-valued function $f : [a,b] \to \mathbb R$ has a minimal and a maximal value. + +> *Proof.* +> - By the boundedness theorem, $f$ is bounded from above. +> - By the completeness of real numbers, a supremum $s$ (least upper bound) of $f$ over $[a,b]$ exists. +> - We want to find $d \in [a,b]$ such that $f(d) = s$. +> +> - As $s$ is the supremum, $s - 1/n$ with $n \in \mathbb N$ is not an upper bound for $f$, thus there exists $d_n \in [a,b]$ with $s - 1/n < f(d_n)$. +> This defines a sequence $(d_n)$. +> +> - By construction +> ```math +> s - \frac1{n} < f(d_n) \leq s. +> ``` +> Hence, $f(d_n) \to s$ as $n \to \infty$. +> +> - As $(d_n)$ is bounded, we obtain from the Bolzano-Weirestrass theorem and the fact that $[a,b]$ is closed, that there exists a converging subsequence $(d_{n_k}) \to d \in [a,b]$. +> - In addition, $f$ is continuous, so we have $f(d_{n_k}) \to f(d)$ as $k \to \infty$. +> +> - As $f(d_{n_k})$ is a subsequence of $f(d_n)$ and $f(d_n) \to s$ as $n \to \infty$, we must have $s = f(d).$ +> +> Again, we can construct an analogous argument for the minimum $\hspace{3.5cm} \square$ +""" + +# ╔═║ 3e81997b-65d5-4e2b-b2c2-95aec0ea10bb +TableOfContents() + +# ╔═║ 2fbc04ff-66e5-41ef-b366-2e54882ee74a +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 330) 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\ No newline at end of file diff --git a/04_Matrix_error_bounds.jl b/04_Matrix_error_bounds.jl new file mode 100644 index 0000000..632528e --- /dev/null +++ b/04_Matrix_error_bounds.jl @@ -0,0 +1,1742 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═║ 8dab1cdf-f700-4dd9-a839-58fe8cfd6c4a +begin + using PlutoUI + import TikzPictures.TikzPicture + using LaTeXStrings + using LinearAlgebra + using PlutoTeachingTools + using Plots + using HypertextLiteral + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ c00bddac-a453-4831-81be-d5814a7e88b9 +md""" +# Bounds on Eigenvalues +""" + +# ╔═║ 440fa8bb-c502-4c51-bd80-b0a86d3b8507 +md""" +Suppose we have computed an approximate eigenpair $(\tilde \lambda, \tilde u)$ of a Hermitian matrix $A \in \mathbb C^{n \times n}$. +- How do we know our computation is correct, in particular if finite-precision arithmetic is employed ? +- How do we know how far away we are from the exact solution ? + +In one of the exercises we already saw that +```math +\begin{align} +\lambda_i \leq \| A \|_F && \forall i = 1, \dots, n +\end{align} +``` +but we also noted it to be a rather crude bound. +This does, however, point to the fact that the matrix entries have something to say about the matrix eigenvalues. + + + + +""" + +# ╔═║ 7eba1e23-9f38-4478-84f4-e81ef653ba2c +md""" +## Gerschgorin circles + +A first tighter bound is given by + +""" + +# ╔═║ 1309b18d-e2f0-46fa-8e0c-3a2730d69d20 +md""" +!!! note "Theorem 1 (Gerschgorin circles)" + Any eigenvalue $\lambda$ of a matrix $A$ is located in one of the closed disks of the complex plane centred at $A_{i i}$ and having the radius + ```math + \sum_{\substack{j=1 \\ j \neq i}}^{j=n} \left|A_{i j}\right| \text {. } + ``` + In other words, + ```math + \forall \lambda \in \sigma(A) \quad \exists\, i\, \text{ s.t. }\, \left|\lambda-A_{i i}\right| \leq \sum_{\substack{j=1 \\ j \neq i}}^{j=n}\left|A_{i j}\right| \tag{1} + ``` + + + +""" + +# ╔═║ 118355a0-2104-4216-9d11-53079fa2024d +md""" +> *Proof* by contradiction. +> - Assume (1) does not hold. Then there is an eigenvalue $\lambda$ such that, for all $i=1, \dots, n$, +> ```math +> \left|\lambda-A_{i i}\right| > \sum_{\substack{j=1 \\ j \neq i}}^{j=n} \left|A_{i j}\right| \text {. } \tag{2} +> ``` +> - We then write +> ```math +> A-\lambda I=D-\lambda I+H, +> ``` +> with $D=\operatorname{diag}\left(A_{11}, \ldots, A_{i i}, \ldots, A_{n n}\right)$ and $H=A-D$ (i.e. zero on the diagonal). +> - Due to (2), $D - \lambda I$ is invertible, thus +> ```math +> A-\lambda I=(D-\lambda I)\Big(I+\underbrace{(D-\lambda I)^{-1} H}_{M}\Big) +> ``` +> - The elements of $M$ are +> ```math +> M_{i j}= \begin{cases}0 & \text{if } i=j \\ \frac{A_{i j}}{A_{i i}-\lambda} & \text {otherwise }\end{cases} +> ``` +> - Due to (2), we thus have $\sum_{j=1}^{n}\left|M_{i j}\right|<1$ $\forall i$, therefore $\|M\|_{\infty}< 1$, and $\|M\|_{2}<\|M\|_{\infty}<1$. +> - Since $\|M\|_{2}$ bounds the modulus of the eigenvalues of $M$, we have that $I+M$ is non-singular, which implies $A-\lambda I$ is non-singular. Therefore $\lambda$ cannot be an eigenvalue of $A$. +> - This contradicts our initial statement. +> $\hspace{7cm} \square$ + +""" + +# ╔═║ 1bae37c2-946f-467e-9b1d-c3ca5a129b98 +md""" +!!! tip "Remark" + Since the same result holds for $A^T$, we can formulate a version in terms of column sums instead of row sums. + ```math + \forall \lambda \in \sigma(A) \quad \exists \,j\, \text { s.t. }\, \left|\lambda-A_{j j}\right| \leq \sum_{\substack{i=1 \\ i \neq j}}^{i = n} \left|A_{i j}\right|. + ``` +""" + +# ╔═║ 9c189c48-497e-43c1-a50e-fc1ebe7f0714 +md""" +The disks defined in this theorem are called **Gerschgorin disks**. There are $n$ disks and their union contains the spectrum of $A$. +Gerschgorin disks are particularly useful when the matrix is almost diagonal, i.e. when a diagonalization algorithm is close to convergence. +""" + +# ╔═║ 371399ec-a571-4aed-b910-7307d90edad1 +md""" +## The residual & Bauer-Fike bound + +Suppose now we have obtained an approximate eigenpair $(\tilde{\lambda}, \tilde{v})$ to $A$. +We can define the **residual** +```math +r=A \tilde{v}-\tilde{\lambda} \tilde{v} +``` +which can be computed and employed as a stopping criterion in iterations. + +Our goal now is to relate the residual $r$ to the error on the eigenvalue, $|\lambda-\tilde{\lambda}|$. +""" + +# ╔═║ 5466b137-915b-4118-9dbf-f97750303784 +md""" +!!! note "Theorem 2 (Bauer-Fike)" + Let $A \in \mathbb{C}^{n \times n}$ Hermitian and let $(\tilde{\lambda}, \tilde{v})$ be an approximate eigenpair with $\|\tilde{v}\|_{2}=1$ and residual $r=A \tilde{v}-\tilde{\lambda} \tilde{v}$. + Then, there exists an eigenvalue $\lambda$ of $A$ such that + ```math + \left|\lambda- \tilde \lambda \right| \leq\|r\|_{2} . + ``` + +> *Proof.* +> - If $\tilde{\lambda} \in \sigma(A)$ the result is trivial. +> - Suppose $\tilde{\lambda}$ is not on eigenvalue of $A$. +> Then $A-\tilde{\lambda} I$ is invertible, thus we write +> ```math +> \begin{align} +> \tilde{v} & =(A-\tilde{\lambda} I)^{-1} r \\ +> & = U \, (D-\tilde{\lambda} I)^{-1} \, U^{-1} \, r \tag{3} +> \end{align} +> ``` +> where in the last step we used that $A$ is Hermitian, thus it can be diagonalised as $A=U D U^{-1}$ with $U$ unitary. +> - Taking the 2-norm on both sids of (3) yields +> ```math +> \begin{aligned} +> 1 =\|\tilde{v}\|_2 & =\left\|U\,(D-\tilde{\lambda} I)^{-1}\, U^{-1}\, r\right\|_2 \\ +> & \leq \underbrace{\|U\|_{2}}_{=1} \ \| (D -\tilde{\lambda} I)^{-1}\|_{2} \ +> {\underbrace{\left\|U^{-1}\right\|_{2}}_{=1}} \, \| r \|_{2} \\ +> & =\max _{i=1,\dots,n} \left|\lambda_{i}-\tilde{\lambda}\right|^{-1} \|r\|_{2} +> \end{aligned} +> ``` +> - Since $\text{argmin}_{i}\, x_{i}=\text{argmax}_i \, x_i^{-1}$, we obtain +> ```math +> \min _{i=1, \ldots, n}\left|\lambda_{i}- \tilde \lambda \right| \leq\|r\|_{2} +> ``` +> as desired. +> $\hspace{12cm} \square$ +""" + +# ╔═║ fd5bcfb5-86c5-4097-a8b0-3f12ef8ed752 +md""" +This is a simple way to get general error bound by establishing a **residual-error relationship**, i.e. a relation between the residual as a computable check for convergence and the error of our quantity of interest against the exact result. +We will note that there is no need to know the exact result ! + +!!! tip "Remark" + In general in *a posteriori* error analysis we want to establish relationship + ```math + \|e\|_{p} \leq C \|r\|_{q} + ``` + where $e$ is the **error against the exact answer**, $r$ the **residual**, + and $C$ a known and computable constant. + **Which norms** $p$ and $q$ are the best choice *depends on context*. + + For example, for measuring the error in the eigenvector we might choose the $\infty$-norm if we are interested in **entry-wise error**, or the 2-norm if we are interested in the error **natural to the vector space** $\mathbb{C}^{n}$. + Note that there is no reason for $q$ or $C$ to be identical in both cases. + +This suggests the following important point: error-residual relationships are not unique. +In fact for our case a better bound is the Kato-Temple bound, which we will derive next. + +## Kato-Temple bound +""" + +# ╔═║ 54fea52c-6adb-4c83-b380-416bfdaeacec +md""" +!!! note "Lemma 3" + Let $\tilde{v}$ be an approximate eigenvector of a Hermitian matrix $A$ with (for simplicity) $\|\tilde{v}\| = 1$. Let $\tilde \lambda$ be the associated approximated eigenvalue, calculated from $\tilde{\lambda}=\langle\tilde{v}, A \tilde{v}\rangle=R_{A}(\tilde{v})$. Further let $(\alpha, \beta)$ be an interval that contains no eigenvalue of $A$, but let $\tilde{\lambda} \in(\alpha, \beta)$. Then + ```math + (\beta-\tilde{\lambda})(\tilde{\lambda}-\alpha) \leq\|r\|_{2}^{2} . + ``` +""" + +# ╔═║ f172a676-599b-4be7-9b5d-ceba6b1794dc +md""" +> *Proof.* +> - First notice $r \perp \tilde{v}$ : +> ```math +> \begin{align*} +> \langle\tilde{v}, r\rangle & =\left\langle\tilde{v}, A \tilde{v}-\tilde{\lambda}\, \tilde{v}\right\rangle \\ +> & =\Big\langle\tilde{v}, A \tilde{v}-\left\langle\tilde{v}, A \tilde{v}\right\rangle \tilde{v}\Big\rangle \\ +> & =\langle\tilde{v}, A \tilde{v}\rangle-\langle\tilde{v}, A \tilde{v}\rangle=0 \tag{4} +> \end{align*} +> ``` +> - Using this result as well as the trick of adding and subtracting $\tilde{\lambda} \tilde{v}$: +> ```math +> \begin{align} +> \Big\langle(A-\alpha I) \, \tilde{v}, (A-\beta I) \, \tilde{v}\Big\rangle +> &=\Big\langle(A-\tilde{\lambda} I)\, \tilde{v} +(\tilde{\lambda}-\alpha)\, \tilde v,\\ +> &\hspace{50pt} +> (A-\tilde{\lambda} I) \, \tilde{v}+(\tilde{\lambda}-\beta)\, \tilde{v} \Big\rangle +> \\ +> & =\Big\langle r+(\tilde{\lambda}-\alpha) \tilde{v},\ r+(\tilde{\lambda}-\beta) \tilde{v}\Big\rangle +> \\ +> & \stackrel{(4)}{=}\|r\|_{2}^{2}+(\tilde{\lambda}-\alpha)(\tilde{\lambda}-\beta). \tag{5} +> \end{align} +> ``` +> - Now expand $\tilde{v}$ in an eigenbasis of $A$, i.e. +> ```math +> \tilde{v}=\sum_{i=1}^{n} c_{i} v_{i} +> ``` +> where $v_i$ is an eigenvector of $A$ with associated eigenvalue $\lambda_i$. +> - This yields for the left-hand side of (5) +> ```math +> \begin{align} +> \big\langle(A-\alpha I) \, \tilde v,\ (A-\beta I) \, \tilde{v}\big\rangle & =\sum_{i=1}^{n}\left|c_{i}\right|^{2}\,(\lambda_i -\alpha)(\lambda_i -\beta) \\ +> & \geq 0 +> \end{align} +> ``` +> since the interval $(\alpha, \beta)$ contains no eigenpairs. +> - Considering the right-hand side of (5), we obtain +> ```math +> \|r\|_{2}^{2}+(\tilde{\lambda}-\alpha)(\tilde{\lambda}-\beta) \geq 0 +> ``` +> which is the desired result. $\hspace{10cm} \square$ + +""" + +# ╔═║ 3ec0e31c-3d09-43d1-9d9a-506320f7d964 +md""" +!!! note "Theorem 4 (Kato-Temple)" + Let $\tilde{v}$ be an approximate eigenvector to $A$, $\|\tilde{v}\|=1, \tilde{\lambda}=\langle\tilde{v}, A \tilde{v}\rangle$, and $r=A \tilde{v}-\tilde{\lambda} \tilde{v}.$ Assume we know an interval $(a, b)$ with $\tilde{\lambda} \in(a, b)$ and where $\lambda$ is the only eigenvalue of $A$ in $(a, b)$. + Then, + ```math + \frac{\|r\|_{2}^{2}}{a-\tilde{\lambda}} \leq \tilde{\lambda}-\lambda \leq \frac{\|r\|_{2}^{2}}{b-\tilde{\lambda}} + ``` +""" + +# ╔═║ a3e89cc9-55b6-423f-874a-b923026847ea +md""" +> *Proof.* +> Let $\lambda$ be the closest eigenvalue to $\tilde \lambda$. +> If $\lambda<\tilde{\lambda}$, take $\alpha=\lambda$ and $\beta=b$ in Lemma 3.3 to yield +> ```math +> \begin{align} +> 0 &\leq(b-\tilde{\lambda})(\tilde{\lambda}-\lambda) \leq\|r\|^{2}_2 \\ +> \Rightarrow \quad 0 &\leq \tilde{\lambda}-\lambda \leq \frac{\|r\|^{2}_2}{b- \tilde{\lambda}} +> \end{align} +> ``` +> Otherwise, set $\alpha=a$ and $\beta=\lambda$ to obtain +> ```math +> 0 \leq \lambda-\tilde{\lambda} \leq \frac{\| r \|^{2}_2}{\tilde{\lambda}-a} +> ``` +> Combining both results completes the proof. +> $\hspace{7cm} \square$ + +""" + +# ╔═║ f9cdf1eb-e803-48be-ab67-cd81fc33be3e +md""" +!!! note "Corollary 5 (Symmetric version of Kato-Temple)" + Let $\tilde{v}$ be an approximate eigenvector to $A$, $\|\tilde{v}\|=1, \tilde{\lambda}=\langle\tilde{v}, A \tilde{v}\rangle$ and $r=A \tilde{v}-\tilde{\lambda} \tilde{v}$. Let $\lambda_i$ be the eigenvalue closest to $\tilde \lambda$. + We define the distance of $\tilde{\lambda}$ to the rest of the spectrum as + ```math + \delta=\min _{j,\, j\neq i} |\lambda_{j}-\tilde{\lambda} | . + ``` + $\delta$ is also sometimes called the **gap**. + Then, + ```math + |\tilde{\lambda}-\lambda| \leq \frac{\|r\|_{2}^{2}}{\delta}. + ``` + +> *Proof.* +> Theorem 3.4 with $a=\tilde{\lambda}-\delta$ and $b=\tilde{\lambda}+\delta$. +""" + +# ╔═║ 4eb5e944-000f-4755-b039-63591d1dab8a +md""" +Note that Corollary 5 does not give a useful bound if $\delta = 0$, which can happen for degenerate eigenvalues. + +Given a sufficiently good eigenvector $\tilde{v}$ **Kato-Temple-type bounds** +are considerably **sharper** than Bauer-Fike bounds. +Their main complication is that the gap $\delta$ requires access to the **exact** eigenvalue and is thus usually not directly computable. +However, one can usually determine an approximate gap as we will detail below. + +- Let us assume a diagonalization routine yields approximate eigenvectors $\tilde{v}_{1}, \ldots, \tilde{v}_{n}$ with eigenvalue approximations $\tilde{\lambda}_{1}, \ldots, \tilde{\lambda}_{n}$ and residuals $\tilde{r}_{1}, \ldots, \tilde{r}_{n}$. We further assume that we have not lost any eigenpair, i.e. $\tilde{Ξ»}_i$ approximates $Ξ»_i$ for all $i = 1, \ldots, n$. +- **A naive idea** to approximate the gap $\delta$ for eigenvalue $\lambda_{i}$ is to take + ```math + \delta_{\text {est}}=\min \left(\left|\tilde{\lambda}_{i-1}- \tilde{\lambda}_{i}\right|,\left|\tilde{\lambda}_{i}-\tilde{\lambda}_{i+1}\right|\right) + ``` + While this *can be* a good approximation, a disadvantage of this approach is + that this expression **is not a guaranteed lower bound** to $\delta$. + In particular it may happen that $\delta_{\text {est}} > \delta$, which implies + that $\left\|r_{i}\right\|^{2} / \delta_{\text{est}}$ + may be smaller than the actual error ! The Kato-Temple estimate is **no longer a guaranteed bound**. + +- To see this pictorially, let us assume for simplicity that all exact eigenvalues are simple. We want to bound $|\lambda-\tilde{\lambda}_{i} |$ by Kato-Temple, and where the exact gap $\delta=\min \left(\left|\tilde{\lambda}_{i}-\lambda_{i+1}\right|,\left|\tilde{\lambda}_{i}-\lambda_{i-1}\right|\right)$. Then it can happen that: +""" + +# ╔═║ c68d3e98-8b17-416b-a231-c5356573580e + + +# ╔═║ f6e33422-d41e-4444-b08b-b281f5eb0db3 +TikzPicture(L""" + \draw[>=latex, ->] (0,0) -- (10,0) node[right]{$\sigma(A)$}; + + \draw[red, dotted] (3,0) -- (3,-1) (5.1,0) -- (5.1,-2) (6.3,-2) -- (6.3,0) ; + \draw[purple, dotted] (3.4,0) -- (3.4,1) (5.1,0) -- (5.1,1) ; + + + \draw (0.5,0) node{$\times$} node[above]{$\lambda_{i-3}$}; + \draw[blue] (0.5 + 0.2,0) node{$\times$} node[below]{$\tilde \lambda_{i-3}$}; + + \draw (1.5,0) node{$\times$} node[above]{$\lambda_{i-2}$}; + \draw[blue] (1.5 + 0.2,0) node{$\times$} node[below]{$\tilde \lambda_{i-2}$}; + + \draw (3,0) node{$\times$} node[above]{$\lambda_{i-1}$}; + \draw[blue] (3 + 0.4,0) node{$\times$} node[below]{$\tilde \lambda_{i-1}$}; + + \draw (4.5,0) node{$\times$} node[above]{$\lambda_{i}$}; + \draw[blue] (4.5 + 0.6,0) node{$\times$} node[below]{$\tilde \lambda_{i}$}; + + \draw (6.3,0) node{$\times$} node[above]{$\lambda_{i+1}$}; + \draw[blue] (6.3 + 1.1,0) node{$\times$} node[below]{$\tilde \lambda_{i+1}$}; + + \draw (8.5,0) node{$\times$} node[above]{$\lambda_{i+2}$}; + \draw[blue] (8.5 + 0.4,0) node{$\times$} node[below]{$\tilde \lambda_{i+2}$}; + + \draw[red] (3,-1) node{|} -- node[below]{$\tilde \lambda_i - \lambda_{i-1}$} (5.1,-1) node{|} ; + + \draw[red] (5.1,-2) node{|} -- node[below]{$\tilde \lambda_i - \lambda_{i+1} = \delta$} (6.3,-2) node{|} ; + + \draw[purple] (5.1,1) node{|} -- node[above]{$\tilde \lambda_i - \tilde \lambda_{i-1} = \delta_{\rm{est}} > \delta $} (3.4,1) node{|}; + + +""",width="23cm",options="scale=1",preamble=raw"\usepackage{amsfonts}") + +# ╔═║ c7c14c64-133f-44e2-83ea-abde368de440 +md""" +- To avoid this problem, we can use Theorem 2 (Bauer-Fike) to obtain a **guaranteed lower bound on the gap**. For example: + ```math + \begin{align} + \left|\tilde{\lambda}_{i}-\lambda_{i+1}\right| & =\left|\tilde{\lambda}_{i}-\tilde{\lambda}_{i+1}+\tilde{\lambda}_{i+1}-\lambda_{i+1}\right| \\ + & \stackrel{\text{rev.}~\Delta}{\geq}\left|\tilde{\lambda}_{i}-\tilde{\lambda}_{i+1}\right|-\left|\tilde{\lambda}_{i+1}-\lambda_{i+1}\right| \\ + & \hspace{-0.5em} \stackrel{\text{Thm 2}}{\geq}\left|\tilde{\lambda}_{i}-\tilde{\lambda}_{i+1}\right|-\left\|r_{i+1}\right\|_{2} + \end{align} + ``` + and similarly for $\left|\tilde{\lambda}_{i}-\lambda_{i-1}\right|$. + +- This results in computable lower bounds to $\left|\tilde{\lambda}_{i}-\lambda_{i+1}\right|$ and $\left|\tilde{\lambda}_{i}-\lambda_{i-1}\right|$, thus $\delta$. + Using this gap estimate thus yields a guaranteed upper bound to $\left\|r_{i}\right\|^{2} / \delta$. + +- Note that when $\lambda_i$ and $\lambda_{i+1}$ are too close, this estimate can be negative, thus not a valid lower bound for $\delta$. +""" + +# ╔═║ 1b5e1cf6-e041-4311-9d9a-1db6cb628a3e +md""" +!!! note "Theorem 6" + Let $\tilde{v}$ be an approximate eigenvector to $A$, $\|\tilde{v}\|=1, \tilde{\lambda}=\langle\tilde{v}, A \tilde{v}\rangle$, and $r=A \tilde{v}-\tilde{\lambda} \tilde{v}$. Let $\lambda$ be the eigenvalue closest to $\tilde{\lambda}$ and $\delta=\min _{i} \{|\lambda_{i}-\tilde{\lambda}|, \lambda_{i} \neq \lambda \}$ be the distance from the spectrum (gap). + Let $v$ be an eigenvector of $A$ associated with $\lambda$. + Recall that $\theta(x, y)$, the angle between two vectors, is defined as + ```math + \cos \theta(x, y)=\frac{|\langle x, y\rangle|}{\|x\|\|y\|}. + ``` + Then, + ```math + \sin \theta(\tilde{v}, v) \leq \frac{\|r\|_{2}}{\delta} + ``` +""" + +# ╔═║ 3c3fabfc-f7e1-4e93-acfa-48c7448f968a +md""" +> *Proof.* +> We set $\theta \equiv \theta(v, \tilde{v})$ and write $\tilde{v}=v \cos \theta+w \sin \theta$, where $w \perp v$ is chosen appropriately. +> We have +> ```math +> \begin{align} +> (A-\tilde{\lambda} I) \tilde{v} & =\cos \theta(A-\tilde{\lambda} I) v +\sin \theta(A-\tilde{\lambda} I) w \\ +> & =\cos \theta({ \lambda} -\tilde{\lambda}) v+\sin \theta(A-\tilde{\lambda} I) w +> \end{align} +> ``` +> Note that +>```math +> \begin{align} +> \langle v, (A-\tilde{\lambda} I) w \rangle & =\langle(A-\tilde{\lambda} I) v, w\rangle \\ +> & =(\lambda-\tilde{\lambda})\langle v, w\rangle=0, +> \end{align} +> ``` +> i.e. the two vectors in the previous sum are orthogonal. +> Therefore +> ```math +> \begin{align} +> \|r \|_{2}^{2} & = \| (A-\tilde{\lambda} I ) \tilde{v} \|_{2}^{2} \\ +> & =\cos ^{2} \theta \ |\lambda-\tilde{\lambda}|^{2}+\sin ^{2} \theta \ \|(A-\tilde{\lambda} I) w\|_{2}^{2} +> \end{align} +> ``` +> Hence, +> ```math +> \sin ^{2} \theta\|(A-\tilde{\lambda} I) w\|_{2}^{2} \leq\|r\|_{2}^{2} +> ``` +> Since $w \perp v$, +>```math +> \begin{aligned} +> \|(A-\tilde{\lambda} I) w\|_{2} & =\left\|\sum_{\substack{i=1 \\ +> \lambda_{i} \neq \lambda}}^{n}\left(\lambda_{i}-\tilde{\lambda}\right) v_{i} v_{i}^{H} w\right\|_{2} \\ +> & \geq \min _{\substack{i=1, \dots, n \\ +> \lambda_i \neq \lambda}}\left|\lambda_{i}-\tilde{\lambda}\right|=\delta +> \end{aligned} +>``` +> This concludes the proof. $\hspace{11cm} \square$ +""" + +# ╔═║ bc4e51aa-674e-4a46-a20d-d9bfd82ff108 +md""" +## Error bounds showcase +""" + +# ╔═║ 45f6befb-4518-4467-81f3-ecd6da88ccbb +md""" +- M12 = $(@bind M12 PlutoUI.Slider(-1.0:0.001:1.0; default=0.001, show_value=true)) +- M13 = $(@bind M13 PlutoUI.Slider(-1.0:0.001:1.0; default=0.1, show_value=true)) +- M14 = $(@bind M14 PlutoUI.Slider(-1.0:0.001:1.0; default=0.1, show_value=true)) +- M23 = $(@bind M23 PlutoUI.Slider(-1.0:0.001:1.0; default=-0.05, show_value=true)) +""" + +# ╔═║ 50024c47-f094-4fed-8ec1-1861a982a49a +M = [ + 1.0 M12 M13 M14 0.0; + M12 2.0 M23 0.0 -0.10; + M13 M23 3.0 0.1 0.05; + M14 0.0 0.1 4.0 0.0; + 0.0 -0.1 0.05 0.0 5.0 +]; + +# ╔═║ ef9e68d5-b5d5-456e-a13d-8c0149c6b2d6 +md""" +Consider the near-diagonal matrix + + +M = $(latexify_md(M)) + + +with some Sliders to tune the off-diagonal elements: +""" + +# ╔═║ a7e14252-02ee-49c1-9f46-5699cf590923 +md""" +- Plot Geschgorin disks: $(@bind show_geschgorin PlutoUI.CheckBox(default=true)) +- Plot Bauer-Fike estimate: $(@bind show_bauer_fike PlutoUI.CheckBox(default=true)) +- Plot Kato-Temple estimate: $(@bind show_kato_temple PlutoUI.CheckBox(default=true)) +""" + +# ╔═║ 2d7cbb67-5c7a-455c-8c19-e9b9e8f4b832 +begin + geschgorin_centres = diag(M) + geschgorin_radii = [sum(abs, row) - abs(row[i]) for (i, row) in enumerate(eachrow(M))] + computed_eigenvalues = diag(M) + exact_eigenvalues = eigvals(M) + computed_eigenvectors = collect(eachcol(Matrix{Float64}(I, 5, 5))) + residuals = map(computed_eigenvalues, computed_eigenvectors) do Ξ», v + M * v - Ξ» * v + end + error_Bauer_Fike = norm.(residuals) + Ξ΄ = map(1:size(M, 2)) do i + Ξ΄_left = Ξ΄_right = Inf + + Ξ»tilde = computed_eigenvalues + if i > 1 + Ξ΄_left = abs(Ξ»tilde[i] - Ξ»tilde[i-1]) - norm(residuals[i-1]) + end + if i < size(M, 2) + Ξ΄_right = abs(Ξ»tilde[i] - Ξ»tilde[i+1]) - norm(residuals[i+1]) + end + max(0.0, min(Ξ΄_left, Ξ΄_right)) + end + error_Kato_Temple = norm.(residuals).^2 ./ Ξ΄ +end; + +# ╔═║ 08508856-57bd-4d17-b8c6-3b0150e5b9af +begin + function make_circle(xmidpoint, radius; n=100) + ΞΈs = (0:n) .* 2Ο€ ./ n + map(radius .* cos.(ΞΈs), radius .* sin.(ΞΈs)) do x, y + x + xmidpoint, y + end + end + + p = plot(; aspect_ratio=1.0, xlims=[0, 6], ylims=[-1.5, 1.5], legend=:bottomright) + scatter!(p, exact_eigenvalues, zeros(size(M, 1)); + label="eigenvalues", mark=:x, markersize=8) + + scatter!(p, computed_eigenvalues, zeros(size(M, 1)); + label="computed", mark=:+, markersize=8) + + for i in 1:size(M, 1) + if show_geschgorin + label = i==1 ? "Gerschgorin" : "" + circle = make_circle(geschgorin_centres[i], geschgorin_radii[i]) + plot!(p, circle; color=3, label, lw=2) + end + + if show_bauer_fike + label = i==1 ? "Bauer-Fike" : "" + circle = make_circle(computed_eigenvalues[i], error_Bauer_Fike[i]) + plot!(p, circle; color=4, label, lw=2) + end + + if show_kato_temple + label = i==1 ? "Kato-Temple" : "" + circle = make_circle(computed_eigenvalues[i], error_Kato_Temple[i]) + plot!(p, circle ; color=5, label, lw=2) + end + end + p +end + +# ╔═║ 760385db-cc30-40c1-8f66-5692028a96e3 +TableOfContents() + +# ╔═║ e5c087db-1df5-4654-a5f3-d4c27a5a9782 +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 265) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +TikzPictures = "37f6aa50-8035-52d0-81c2-5a1d08754b2d" + +[compat] +HypertextLiteral = "~0.9.5" +LaTeXStrings = "~1.3.1" +Plots = "~1.40.5" +PlutoTeachingTools = "~0.2.15" +PlutoUI = "~0.7.58" +TikzPictures = "~3.5.0" +""" + +# ╔═║ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "dae857aa02f1b28b551de5211af7daea92cb0cbe" + +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "6e1d2a35f2f90a4bc7c2ed98079b2ba09c35b83a" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.3.2" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.BitFlags]] +git-tree-sha1 = "0691e34b3bb8be9307330f88d1a3c3f25466c24d" +uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35" +version = "0.1.9" + +[[deps.Bzip2_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 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"libaom_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"] +git-tree-sha1 = "74faea50c1d007c85837327f6775bea60b5492dd" +uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5" +version = "4.4.2+2" + +[[deps.FileWatching]] +uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee" + +[[deps.FixedPointNumbers]] +deps = ["Statistics"] +git-tree-sha1 = "05882d6995ae5c12bb5f36dd2ed3f61c98cbb172" +uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93" +version = "0.8.5" + +[[deps.Fontconfig_jll]] +deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Zlib_jll"] +git-tree-sha1 = "db16beca600632c95fc8aca29890d83788dd8b23" +uuid = "a3f928ae-7b40-5064-980b-68af3947d34b" +version = "2.13.96+0" + +[[deps.Format]] +git-tree-sha1 = "9c68794ef81b08086aeb32eeaf33531668d5f5fc" +uuid = "1fa38f19-a742-5d3f-a2b9-30dd87b9d5f8" +version = "1.3.7" + +[[deps.FreeType2_jll]] +deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Zlib_jll"] +git-tree-sha1 = 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\ No newline at end of file diff --git a/05_Floating_point_arithmetic.jl b/05_Floating_point_arithmetic.jl new file mode 100644 index 0000000..451bc61 --- /dev/null +++ b/05_Floating_point_arithmetic.jl @@ -0,0 +1,2879 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═║ fcf3b117-bb14-4849-818c-a0dfc4de15aa +begin + using PlutoUI + using Plots + using PlutoTeachingTools + using HypertextLiteral +end + +# ╔═║ 3a59c0c6-5496-4faf-ae89-3c6333d02abb +using BFloat16s + +# ╔═║ baa1625c-7faa-4341-a0b6-bb5ad3fdb67a +using LinearAlgebra + +# ╔═║ 3e0b819d-9dbe-45f7-9e92-c58578b85d21 +using AccurateArithmetic + +# ╔═║ a06746c0-c3c3-46cf-985f-2d6873f1f141 +using BenchmarkTools + +# ╔═║ 67d5a7a0-9866-41bf-a242-70d9ae44a444 +begin + using DoubleFloats + using GenericLinearAlgebra +end + +# ╔═║ 8da0c6e6-ba1a-41ae-99f1-060a6cb53bbc +using IntervalArithmetic + +# ╔═║ a3d727fa-fce6-43c7-ac54-51b5bcc0cde0 +md""" +# Tackling errors due to floating-point arithmetic +""" + +# ╔═║ b1c739ed-957b-4a57-b1aa-6981bb3f76c4 +md""" +In this lecture we will discuss some properties of floating-point numbers and floating-point arithmetic. How are numbers represented on the computer? What can we say about the error introduced by rounding? Are there computational tools to estimate or reduce floating-point error in our computation? + +To illustrate our points we will dwell from a number of simple problems, not all related to eigenvalue computations per se. + +This material is partly taken from the first 3 chapters of the excellent book *Accuracy and Stability of Numerical Algorithms* by Nicholas J. Higham and a recent Acta Numerica paper titled [Floating-point arithmetic](https://doi.org/10.1017/S0962492922000101) by Boldo, Jeannerod, Melquiond and Muller. + +A good resource is also the [Julia documentation](https://docs.julialang.org/en/v1/manual/integers-and-floating-point-numbers/) on this topic. +""" + +# ╔═║ 3c06f892-29d9-4df6-bd85-b22a19d12350 +TableOfContents() + +# ╔═║ 77c9a481-b314-42f3-944e-b6422b894dd0 +md""" +## Basic concepts +### Some notation and definitions + +- Our main measure of error are the **absolute error** + ```math + e_\text{abs} = |x - \hat{x}| + ``` + and the **relative error** + ```math + e_\text{rel}(\hat{x}) = \frac{|x - \hat{x}|}{|x|}. + ``` + For scientific computations the order of magnitude of an expression can vary a lot and thus typically the relative error is of main importance. Notice, that an equivalent way to express the relative error is + ```math + e_\text{rel}(\hat{x}) = |Ξ΄| \qquad \text{where} \qquad \hat{x} = x (1 + Ξ΄) + ``` +- Notice that the relative error is related to the number of correct **significant digits**, but generally considering the relative error instead of the notion of significant digits is clearer. In the following examples, $x$ and $y$ agree with their respective approximation to 3 significant digits, but the relative error varies largely: +""" + +# ╔═║ 0e5d0819-d101-4492-8bb0-73d5302ce461 +let + x = 1.00000 + xhat = 1.00499 + E_rel = abs(x - xhat) / abs(x) +end + +# ╔═║ ac79b644-449d-4684-bc14-f06ca24ee160 +let + y = 9.99999 + yhat = 9.99899 + E_rel = abs(y - yhat) / abs(y) +end + +# ╔═║ fd0a3bea-2178-4df5-96f4-b24a7e46b512 +md""" +- The function $fl(\,\cdot\,)$ denotes evaluation of an expression using floating-point arithmetic. For basic arithmetic operations, such as $+$, $-$, $\ast$, $/$ we will assume + ```math + fl(x\, \text{op}\, y) = (x\, \text{op}\, y) (1 + Ξ΄) \qquad |Ξ΄| < u + ``` + where $u$ is the **unit roundoff** (or machine precision). In other words we assume that $u$ is an upper bound to the relative error of all basic operations. This is the so-called **standard model** of floating-point arithmetic. +- **Computed quantities** will be denoted by a hat, i.e. $\hat{x}$ is the floating-point approximation to $x$. +- Notice the difference between **precision** and **accuracy**: Whereas *accuracy* refers to the *relative error of an approximate computation* versus the exact result, *precision* refers to the accuracy with which basic operations are performed, which itself is bounded by the unit roundoff $u$ as discussed above. Most importantly precision is not an upper bound to accuracy. We will discuss techniques where computations are done in double precision arithmetic, but more accuracy than double precision is achieved. +""" + +# ╔═║ cba52909-94af-4185-a990-36583976643c +md""" +### Forward and backward error + +Assume we have a computation $y = f(x)$, where $f$ is a function $\mathbb{R} \to \mathbb{R}$. Our computation yields a result $\hat{y}$. How should we measure the "quality" of $\hat{y}$? Given that we represent both our computation and our answer in a floating-point number system of precision $u$ we are probably happy with an answer $e_\text{rel}(\hat{y}) \leq u$. However, this is hardly ever feasible. + +A slightly different point of view is to take a look on the input data $x$. Given an approximate answer $\hat{y}$, for what range of input data do we actually obtain this solution? Or in other words for which $\Delta x$ do we have $\hat{y} = f(x + \Delta x)$. Notice that in general there are many such answers $\Delta x$, and we are typically interested in obtaining bounds on the largest $|\Delta x|$, which achieves $\hat{y} = f(x + \Delta x)$. This process is called **backwards error analysis** and $|\Delta x|$ is referred to as **backwards error**. To distinguish more clearly the errors in $\hat{y}$, we refer to these as **forward errors**. + +An algorithm is called **backwards stable** if for any $x$ it produces a computed $\hat{y}$ with a small backwards error, that is $\hat{y} = f(x + \Delta x)$ is only achieved for a small $\Delta x$. What small means depends on context. + +Let us consider an example to illustrate these concepts further. We consider the basic operation +```math + y = f(x) = x - 1.0 +``` +""" + +# ╔═║ 21d26883-5dcb-409b-bbee-64c6155b4d53 +f(x) = x - 1.0 + +# ╔═║ 2f7c1a64-bee2-470f-a2f6-887fad89994f +md""" +Based on our assumption of the standard model, the operation $x-1.0$ will yield the exact answer for any perturbed data $x (1 + Ξ΄)$ with $|Ξ΄| < u$. By assumption this operation is backwards stable. We can test this numerically for example for the case $x = 1 + 10^{-16}$. In double precision +""" + +# ╔═║ 408f1b1c-dd98-4e74-92ad-754b6122e525 +md"""Since the result is $\hat{y} = 0.0$ (instead of the exact $10^{-16}$) $f$ has a forward error of $10^{-16}$ in this example (which is less than $u$). Now""" + +# ╔═║ 4751e59e-98fd-4547-bcd1-43ae740e2e6f +f(1.0) + +# ╔═║ 6003c907-3722-4fd0-9eae-ab96ec88d902 +md"thus taking a $\Delta x = -10^{-16}$ yields the same as $f(x)$ for our $x=1+10^{-16}$. +Further for $\Delta x = 2 \cdot 10^{-16}$" + +# ╔═║ fa2932d3-b09b-4ff2-8d59-295c4fe8cd5d +md""" +Therefore the backward error $|\Delta x|$ is between $10^{-16}$ and $2 \cdot 10^{-16}$ in this example. + +Following further along the idea of errors of floating-point addition, we find that the first number larger than $1$, which is representable in `Float64` is +""" + +# ╔═║ 6f7058fe-171a-46a9-9d07-ccd65fb61bcd +md""" +where `eps` gives the **unit in the last place** (ulp) of its input, which is defined as +""" + +# ╔═║ 740d9e34-71c0-46dc-a9a0-0e4537e63e69 +epsdef(x) = max(x - prevfloat(x), nextfloat(x) - x) + +# ╔═║ 8d126d65-34b3-478b-a1c8-03142fa4e888 +md""" +Where `prevfloat` and `nextfloat` return the previous and next representable floating-point number to a given number, e.g. +""" + +# ╔═║ 1cb60604-7d36-4f11-9683-d5a4ca821c7e +prevfloat(1.2345) + +# ╔═║ 2ec1b321-c63d-40d3-a00b-49b7a80098f8 +nextfloat(-1e-2) + +# ╔═║ 293e2d97-5e6f-4d69-99f9-1a9dde7e1c30 +md""" +Due to its importance, `eps(one(T))` is also called the **machine epsilon** of the floating-point type `T`. For example: +""" + +# ╔═║ c0d6e714-8c25-45eb-afe8-3fae25a174e6 +eps(Float32) + +# ╔═║ 9368e779-9104-44e9-971d-95bc30ee38bc +eps(Float64) + +# ╔═║ 8b261d9b-22e3-4596-971d-30e9f51a6a0f +md""" +### Conditioning and condition number + +Forward and backward error can be related to each other in most cases. The relationship between them is usually called the **conditioning**, that is the sensitivity of the solution to the perturbation of the input data for a numerical scheme. + +Consider our previous example with $y = f(x)$ and $\hat{y} = f(x + \Delta x)$. Then if $f$ is differentiable we can find an expansion +```math +\hat{y} - y = f(x + \Delta x) - f(x) = f'(x) \Delta x + O\big( (\Delta x)^2 \big). +``` +Dividing by $y = f(x)$ then gives +```math +\frac{\hat{y} - y}{y} = \left( \frac{x f'(x)}{f(x)} \right) \frac{\Delta x}{x} + O\big( (\Delta x)^2 \big), +``` +which relates the relative forward error to the relative backward error by means of the factor +```math +\kappa = \left| \frac{x f'(x)}{f(x)} \right| +``` +the so-called **condition number**. + +When defined consistently, the following useful relationship holds between forward/backward error and condition number: +```math +\text{forward error} \lessapprox \text{condition number} \times \text{backward error}. +``` +Crucially, this implies that for an ill-conditioned problem, a small backward error does *not* imply we have solved the problem to high accuracy, meaning that the reminiscent forward error may still be rather large. +In other words even if we employ a backwards-stable algorithm to compute our approximate solution $\hat{y}$ and $\Delta x$ is small our relative error in $\hat{y}$ may still be large. + +For iterative methods this implies for ill-conditioned problems: Even if successive iterates $x^{(i)}$ and $x^{(i+1)}$ are very close and our algorithm might not even be able to improve any more (e.g. $|x^{(i+1)} - x^{(i)}| < u$), still the error in the solution $\hat{y}$ we might still be large. +""" + +# ╔═║ ab77c84b-c0a3-436b-ac49-cd667983ead8 +md""" +**Vector functions.** +This argument can be generalised to functions between vectors by taking vector norms to compute (relative) errors and considering the *maximal* possible ratio between forward and backward error. +For example, we want to solve the **linear system** $Ay = x$, which can be done by the solution "algorithm" $f(x) = A^{-1} x$, which clearly is a function $\mathbb{R}^n \to \mathbb{R}^n$ with $n$ the dimension. +If we are able to perturb $x$ to $x + \Delta x$ to obtain the computed $\hat{y}$, +the relative forward error is +```math +e_\text{fw} = \frac{\| \hat{y} - y \|}{\| y \|} = \frac{\| A^{-1} (x + \Delta x) - A^{-1} x \|}{ \| A^{-1} x \|} = \frac{\| A^{-1} \Delta x \|}{ \| A^{-1} x \|} +``` +and the relative backward error clearly +```math +e_\text{bw} = \frac{ \| x + \Delta x - x \| }{\| x \|} = \frac{\| \Delta x \|}{\| x \|}, +``` +Taking the maximal element of the ratio between the expressions over all possible non-zero values of $\Delta x$ and $x$ we obtain the relative condition number as +```math +\begin{align} +\kappa &= \max_{x\neq 0} \max_{\Delta x \neq 0} +\frac{e_\text{fw}}{e_\text{bw}} \\ +&= \max_{x\neq 0} \max_{\Delta x \neq 0} +\frac{\| A^{-1} \Delta x \| \| x \| }{\| A^{-1} x \| \| \Delta x \| } \\ +&= \max_{x\neq 0}\frac{\| x \|}{\| A^{-1} x \| } \max_{\Delta x \neq 0} +\frac{\| A^{-1} \Delta x \| }{ \| \Delta x \| } \\ +&= \max_{y\neq 0}\frac{\| A y \|}{\| y \| } \max_{\Delta x \neq 0} +\frac{\| A^{-1} \Delta x \| }{ \| \Delta x \| } \\ +&= \| A \| \, \| A^{-1} \|, +\end{align} +``` +which is the familiar expression for the condition number of linear systems. + +Notice: For eigenvalue problems the condition number takes a different expression. A detailed discussion is deferred to our discussion of perturbation theory in eigenvalue problems. +""" + +# ╔═║ 3edfa209-d265-4b90-9461-a2d0a2461a5f +md""" +### Cancellation + +Subtractive cancellation happens when nearly equal numbers are subtracted. This is often (but not always) a bad thing. We consider the function +""" + +# ╔═║ bc5e8449-bf43-4c4e-8d12-bf035e1413f7 +fαΆœα΅’Λ’(x) = (1 - cos(x))/x^2 + +# ╔═║ 102f8541-11bd-4b91-a8bd-21acb683ae20 +md""" +By considering the Taylor expansion of `cos` one easily shows that +$0 \leq f^\text{cos}(x) < 1/2$ for all $x \neq 0$. However, we obtain +""" + +# ╔═║ 7f21c4b2-6ac0-4bd4-88ee-5a48fa3be773 +fαΆœα΅’Λ’(1.2e-8) + +# ╔═║ e61319a9-2a02-452e-96ac-60bc35f12a32 +md""" +which is thus clearly wrong ... not even the first digit is correct. The culprit here is the computation in the numerator of this function, since +""" + +# ╔═║ 360b47ab-992c-43a0-9afa-61865a3352ab +c = cos(1.2e-8) + +# ╔═║ 7400e423-da76-4a61-b32e-bbb08c93609d +md"such that" + +# ╔═║ a7f2a54b-e977-4c6f-a24d-994019d57b4b +1 - c + +# ╔═║ a4d4ced6-b71d-4fd2-8a26-262a443bc6c3 +md"""which only has 1 significant figure left. +It is worth emphasising that the subtraction `1 - c` is actually *exact*. The error has been introduced in the computation of `c` itself. In other words subtractive cancellation *brings earlier errors into prominence*. + +To analyse this we consider the subtraction of two computed numbers $\hat{a}$ and $\hat{b}$ in *exact arithmetic* to form $\hat{x} = \hat{a} - \hat{b}$. Denoting by $\Delta a$ and $\Delta b$ the relative errors of $\hat{a}$ and $\hat{b}$ from previous computations, i.e. +```math +\hat{a} = a (1 + \Delta a) \qquad \hat{b} = b (1 + \Delta b) +``` +then we have (using $x = a - b$): +```math + \left| \frac{x - \hat{x}}{x} \right| += \left| \frac{-a \Delta a + b \Delta b }{a - b} \right| +\leq \max(|\Delta a|, |\Delta b|) \frac{|a| + |b|}{|a - b|}. +``` +Important to note is that the relative error bound in $\hat{x}$ is thus in particular large when $|a - b| \ll |a| + |b|$, i.e. when we have heavy cancellations in the subtraction. Note that cancellation is not *always* a bad thing. In case $a$ and $b$ are exact (i.e. error-free) the subtraction is actually exact (in the exact arithmetic we have assumed here) or its error bounded by the usual $u$ (if the standard model of FP arithmetic applies). +""" + +# ╔═║ 399a584e-5c21-4502-b89e-ea4a3871a3f8 +md""" +### Rounding errors + +When performing floating-point operations rounding is inevitable to bring computed values back to the valid range of the employed floating-point datatype. While rounding in the IEEE standard has been designed to be reasonably balanced, such that one may expect errors to cancel or evenly distribute (and they often are), if one is not careful a few rounding errors in an iterative procedure can lead to rather catastrophic results. But in fact rounding errors are not just bad as we will see. + +**Accumulation of rounding errors.** +As an illustration consider taking finite $n$ in the definition +```math +e = \lim_{n\to\infty}\left(1 + \frac1n\right)^n +``` +defined in the function +""" + +# ╔═║ bfff1a42-d18c-4bdb-bf3f-224ba5261eef +md""" +where `T` varies the employed floating-point type. We plot for a few floating-point types: +""" + +# ╔═║ 18754b1e-f8d5-43ee-a668-a13dcd19f2e0 +md""" +Clearly the approximation is poor and moreover degrades completely as $n$ approaches the reciprocal of machine precision. + +The reason is that in a binary floating-point system fractions like $1/10^n$ are non-terminating numbers, e.g. +""" + +# ╔═║ 188c06ba-2201-40ec-b14f-042da0c12444 +bitstring(1.0 / 10.0) # Shows the binary representation of a variable as a string + +# ╔═║ c19abeaa-7e9f-4a1f-892a-43ef9c19ee1a +md""" +as a result when $1 + 1/n$ is formed for $n$ being large powers of $10$, only a few significant digits of $1/n$ are retained in the sum. The subsequent exponentiation must thus produce an inaccurate approximation to $e$, even if done exactly. + +An important takeaway of this example is that a **single rounding error** was enough to cause this behaviour. No accumulation and successive rounding is involved. +""" + +# ╔═║ e8d93fd9-e32b-476f-9468-e637089a625f +md""" +**Rounding errors can be beneficial.** +It is important to keep in mind that rounding errors (or in general floating-point errors) are not just problematic for numerical algorithms. They can also be beneficial. To see this we return to our good old friend the Power method. An implementation we used a few times is as follows: +""" + +# ╔═║ 39e0ed4f-c812-47dd-b55d-3c4b5e07b2f5 +function power_method(A, u=randn(eltype(A), size(A, 2)); + tol=1e-8, maxiter=100, verbose=true) + norm_Ξ”u = NaN + for i in 1:maxiter + u_prev = u + u = A * u + normalize!(u) + norm_Ξ”u = min(norm(u - u_prev), norm(-u - u_prev)) + norm_Ξ”u < tol && break + verbose && println("$i $norm_Ξ”u") + end + ΞΌ = dot(u, A, u) + norm_Ξ”u β‰₯ tol && verbose && @warn "Power not converged $norm_Ξ”u" + (; ΞΌ, u) +end + +# ╔═║ 3f83b876-54a6-4900-be2a-d63609ed1947 +md"""We consider the matrix""" + +# ╔═║ b822af3d-36e9-4985-a85f-eb34f96a660c +A = [0.4 -0.6 0.2; -0.3 0.7 -0.4; -0.1 -0.4 0.5] + +# ╔═║ 2361d15d-01c9-4e5d-98c4-ddec8ec2a19a +md"""This matrix has eigenpairs""" + +# ╔═║ 5d571dcf-aef0-448e-8eaf-704d4b02b124 +Ξ», X = eigen(A) + +# ╔═║ ef685429-5b8d-43ea-9480-eb2791345a9a +md"""With in particular the eigenvector (corresponding to the zero eigenvalue)""" + +# ╔═║ fd98ab89-275c-49b4-9a46-17880f2b6540 +vstart = [1.0; 1.0; 1.0] + +# ╔═║ 53dec758-3c39-4d2c-b9f4-58dba5b45c68 +md""" +being orthogonal to the eigenvector of the largest eigenvalue of $A$. In exact arithmetic the power method would thus converge in one step --- namely to the eigenvalue zero and the normalised starting vector. + +However, we observe +""" + +# ╔═║ 3edb0d0f-4b75-4fa5-a5e3-80129abe2220 +begin + res = power_method(A, vstart) + res.ΞΌ +end + +# ╔═║ 1b4bc39a-eed6-451a-b16c-fb5025ff7012 +md""" +... in other words a nice convergence to the correct largest eigenvalue and corresponding eigenvector. + +Interestingly the reason why this works is exactly the reason why our previous example failed, namely the fact that decimal fractions (like $-0.1$ or $0.2$) are not exactly representable in binary floating-point formats. We actually diagonalise $A+Ξ”A$, where $Ξ”A$ is a tiny perturbation. For $A+Ξ”A$ `vstart` is not an exact eigenvector and thus the overlap with its largest eigenvector is non-zero, and the iteration can proceed. Note, that the final eigenpair we obtain is thus similarly an approximate eigenpair to $A+Ξ”A$ and not $A$, but in fact due to $Ξ”A$ being small the eigenpairs of $A+Ξ”A$ are very good approximations to those of $A$. +""" + +# ╔═║ 5c081511-7391-4d83-a1cf-ff83342b856e +md""" +**Rounding errors are not random.** +This simple fact is indeed exploited in many stable algorithms, which sometimes even heavily rely on the cancellation of rounding errors. While there are attempts to construct approximate statistical models to characterise rounding errors in computations, there are a number of illustrative examples, where rounding errors follow clear patterns. One example (due to W. Kahan) is the rational function +""" + +# ╔═║ 96e1effa-d45e-4b59-b019-93b08b12dca3 +r(x) = (622 - x * (751 - x * (324 - x * (59 - 4x)))) / + (112 - x * (151 - x * ( 72 - x * (14 - x)))) + +# ╔═║ ec14869d-e277-4628-b1ea-3823db0a2a0f +md"""which is expressed in a form how a ratio of quartic polynomials would be evaluated using Horner's method. Horner's method is a very efficient and thus frequently employed way to evaluate rational functions. For example the evaluation of many standard functions such as `sin`, `cos`, `erf`, etc. are under the hood evaluated using Horner's method. + +We evaluate `r(x)` for the numbers""" + +# ╔═║ e19d8e1b-7e75-4e76-baa6-4d2c5bc15496 +md"""A striking pattern due to the rounding can be observed.""" + +# ╔═║ 8c32c50a-82aa-4750-b91f-e393e2eab11e +md""" +### Increasing the precision to check accuracy + +A common technique for estimating the accuracy of a computed result is to redo the computation with increased precision and check if the presumably more accurate answer agrees. Many algorithms --- especially if the main source of error is rounding --- do possess an error bound that decreases when the precision is increased. This implies that at infinite precision, the exact answer is obtained. However, there is generally no guarantee that increasing the precision *always* improves the answer. + +To illustrate this point numerically, we exploit that the `BigFloat` type in julia actually allows to dynamically modify the number of bits that are employed for the underlying computation: +""" + +# ╔═║ 31be711b-4250-49e8-b9d5-3cf4db45108b +md""" +We observe that over the full range between 22 and 31 bits the error more or less stagnates, only to decrease later on. + +Thus: If **no significant change** in the result is obtained when repeating a calculation **at higher precision**, this is **not a sign we got the right answer**! + + +""" + +# ╔═║ 97db876f-2aca-45fa-8b05-229ac1139a80 +md""" +### Common misconceptions about floating-point errors + +In this first part of the lecture we provided a few examples illustrating the difficulties of floating-point operations. We want to pause for a summary of common misconceptions, some of which we disproved by numerical examples shown above: + +1. **Cancellation in the subtraction of numbers is always a bad thing.** No: It sometimes leads to convergence to the best possible answer. +2. **Rounding errors can only overwhelm a calculation if a vast number of them accumulate.** No: Sometimes a single rounding error is sufficient. +3. **Increasing the precision always improves the accuracy of the final answer.** No: Stagnation or even deterioration of the answer can occur. +4. **The final answer of an algorithm is always less accurate than its intermediate quantities.** No: Stable algorithms are often even design around the cancellation of rounding errors, for example. +5. **Rounding errors can only hinder, not aid the success of a computation.** No: See the Power method example. +""" + +# ╔═║ 75bed4a9-54f5-44c1-bcc3-74d5fb11c88a +md""" +## Floating-point number systems +Overall our discussion above (e.g. the aspects of rounding errors) shows, how tricky it is to design a good floating point system, keeping all the beneficial aspects of finite-precision arithmetic, but reducing the impact of rounding, cancellation with respect to the final accuracy obtained. +""" + +# ╔═║ 1d5c2471-1343-4706-9894-293da149ae74 +md""" +**Structure of floating-point numbers.** +A floating point number system $F \subset \mathbb{R}$ has elements of the form +```math +y = \pm m \, \beta^{e-t}. +``` +It is thus characterised by four integer parameters: +- the **base** (or radix) $\beta$ (typically binary, i.e. $Ξ² = 2$), +- the **precision** $t$ and +- the **exponent range** $e_\text{min} \leq e \leq e_\text{max}$. + +In the definition of $y$ the integer $m$ is the **significand** (or mantissa), which satisfies $0 \leq m \leq Ξ²^t - 1$. + +To ensure a *unique* representation for each non-zero $y \in F$, we assume $m \geq Ξ²^{t-1}$. Such numbers are called **normalised**. The $0$ is a special case, which does not have a normalised representation. + +**Example:** To illustrate some characteristics of floating-point numbers, we consider an example system with $\beta = 2$, $t = 3$, $e_\text{min} = -1$ and $e_\text{max} = 3$. Considering only the positive numbers, the following numbers can be represented: +""" + +# ╔═║ 71e0fa21-4cfc-45a2-b6d4-a703a892c15d +md"Plot limits: $(@bind plot_xlim PlutoUI.Slider([0.5, 1.0, 2.0, 4.0, 8.0]; default=8.0))" + +# ╔═║ a9279bfa-7014-499d-9a64-0f6550a665f8 +md""" +We can read off from the plot: +- The **range** of non-zero floating-point numbers is $Ξ²^{e_\text{min} - 1} \leq |y| \leq Ξ²^{e_\text{max}} (1 - Ξ²^{-t})$. +- The spacing of floating-point numbers jumps by a factor of $2$ each power of $2$ and thus is clearly not even. This spacing is characterised by the **machine epsilon** (distance to next larger floating-point number at $1.0$), which is $\varepsilon_M = Ξ²^{1-t}$. $\varepsilon_M$ is the spacing between numbers in the range $[1.0, Ξ²]$, whereas the spacing in the interval $[1/Ξ², 1]$ is $Ξ²^{-t} = \varepsilon_M/Ξ²$. + +In fact one can prove *(Higham, Lemma 2.1)*: + +> The spacing between a normalised floating-point number $x$ and an adjacent normalised floating-point number is at least $Ξ²^{-1} \varepsilon_M |x|$ and at most $\varepsilon_M |x|$. + +The number system $F$ can be extended by including so-called **subnormal numbers**, that is numbers of the form +```math +y = \pm m \, Ξ²^{e_\text{min} - t}, \qquad 0 < m < Ξ²^{t-1}, +``` +i.e. which have the minimum exponent and are *not* normalised. These numbers have fewer digits of precision, but are useful to fill the gap between the smallest positive normalised number $\lambda = Ξ²^{e_\text{min} - 1}$ and $0$. Note, that the smallest positive subnormal number is $\lambda \varepsilon_M = Ξ²^{e_\text{min} - t}$. +Adding these to our example above we obtain (note the different x-axis scale): +""" + +# ╔═║ 0be06f95-8464-479e-8b2c-3840669354b8 +md""" + +**Rounding.** Having the structure of the floating-point numbers in place, we now consider **rounding**, i.e. the transformation $x \mapsto fl(x)$, which maps $\mathbb{R}$ to numbers of the form $\pm m \, \beta^{e-t}$ (which could be subnormal). Rounding is monotonic, i.e. $x \leq y$ implies $fl(x) \leq fl(y)$. We say that $fl(x)$ **overflows** if $|fl(x)| > \max\{|y| \, | \, y\in F\}$ and we say it **underflows** if $0 < |fl(x)| < \min\{|y| \, | \, 0\neq y\in F \}$. + +In practice different ways of rounding are employed, see the Julia docstring of the `RoundingMode` structure. If the rounding mode does not matter we use $fl$ to denote rounding, if it does we employ the more specific notation: + - `RoundNearest` in Julia or $RN(\,\cdot\,)$: The IEEE default rounding mode, which rounds to the closest representable number in $F$ with ties being round to the value, which has a `0` bit in the least significant bit of $m$. + - `RoundUp` in Julia or $RU(\,\cdot\,)$: Always round up to the next largest representable number. + - `RoundDown` in Julia or $RD(\,\cdot\,)$: Always round down to the next smallest representable number. + +With this setup one can prove rigorously that every real number that lies within the range of $F$ can be approximated by $F$ with an error no larger than the **unit roundoff** $u = \frac12 Ξ²^{1-t} = \frac12 \varepsilon_M$, namely *(Higham, Theorem 2.2 and Theorem 2.3)*: + +> If $x \in \mathbb{R}$ lies within the range of $F$ then +> ```math +> \begin{aligned} +> fl(x) &= x (1 + Ξ΄), &&|Ξ΄| < u \\ +> fl(x) &= \frac{x}{1 + \tilde{Ξ΄}}. && |\tilde{Ξ΄}| \leq u +> \end{aligned} +> ``` + +This key result and the assumption of the *standard model* (see Basic notation subsection above) are the main ingredients underlying the numerical analysis of floating-point arithmetic. +""" + +# ╔═║ 1dcdb5e6-a4a0-40a9-9cf0-3072bad872f4 +md""" +### IEEE 754 and quasi-standard formats + +Until the mid-1980s floating-point numbers where not standardised. This meant that pretty much all major manufacturers adopted different solutions for representing numbers. Details like the base used to represent numbers, the ways of rounding computations to representable numbers or how to handle exceptional cases such as forbidden operations, overflow, underflow were *different* on every machine. Imagine writing portable software for this mess .... Clarity finally brought the IEEE Standard 754, which was originally published in 1985 and has been revised a few times since. + +The standard notably defines not only the floating-point formats and some standard rounding modes, but also floating-point exceptions and their handling as well as conversions between integers and floats or floats and other formats. + +The main focus of the standard has been on *high-precision* computation. However, in more recent applications such as statistical learning the reduced memory requirement of lower-precision floating-point formats is nowadays a key ingredient to obtain peak performance. This has lead to the introduction of a few **non-standard $16$-bit formats**, which have become widespread and *de facto* standards. The main differences of these formats compared to IEEE is their tradeoff between the number of bits employed for representing the significand and the exponent. + +Key parameters of these formats are summarised in the following table: + +Type | Size | Significand bits $t$ | Exponent bits | $e_\text{min}$ | $e_\text{max}$ | Unit roundoff $u$ +:---- | :---- | :--: | :------: | -------------: | -------------: | :------------- +`Bfloat16` | 16bit | 7+1 | 8 | -127 | +126 | $2^{-8} \simeq 0.004$ +`Float16` | 16bit | 10+1 | 5 | -15 | +14 | $2^{-11} \simeq 5\cdot 10^{-4}$ +`Float32` | 32bit | 23+1 | 8 | -127 | +126 | $2^{-24} \simeq 6\cdot 10^{-8}$ +`Float64` | 64bit | 52+1 | 11 | -1023 | +1022 | $2^{-53} \simeq 1\cdot 10^{-16}$ + +All formats discussed here are binary (i.e. $Ξ² = 2$) and further enforce that all numbers are normalised or subnormal. As a result the most significant bit of the significand is always $1$ and thus not stored. This is indicated by the $+1$ in the table above. + +Note, that in Julia all aforementioned floating-point types are directly supported. Only `Bfloat16` requires an additional package: +""" + +# ╔═║ f2752b0d-b1d3-4f3d-bb0c-192177a4f997 +u_Bfloat16 = eps(BFloat16) / 2 # Compute the u for BFloat16 + +# ╔═║ 0fb4ef70-e529-41bc-a00d-3095d21d8ea2 +md""" +**IEEE arithmetic is closed:** The standard requires every operation on an IEEE number to produce a result, which is an IEEE number --- whether this makes mathematical sense or not. Exceptional operations raise a signal, which by default is handled by setting a flag (e.g. in form of a special bit pattern). +For example: +""" + +# ╔═║ a7f9953b-0a14-4d6c-bed6-63a826d53a08 +floatmax(Float64) * 10.0 # Overflow + +# ╔═║ 2c625ae1-0275-4548-b8c1-a26a8ba50794 +10.0 / -0.0 # Divide by zero + +# ╔═║ 64b720d1-c38d-485e-9243-c501c9155636 +0.0 / 0.0 # Invalid operation + +# ╔═║ 2901a2eb-fc06-42e8-90fd-b3932529bc60 +Inf / -Inf # Invalid operation + +# ╔═║ 1c4483d3-37f4-4043-93f6-a3ba09bcac36 +md"""Note that `NaN` is **not a number** and thus compares as false to everything, including itself:""" + +# ╔═║ 889f1186-de4e-4060-91b1-bc2511c7828e +NaN == NaN + +# ╔═║ 6ffe44f7-e507-4bd2-bf30-b06ccb0b2027 +md"Notably the zero can be signed to keep sign information in case of underflow:" + +# ╔═║ 35466aab-f3ba-4953-8fae-d545bc218ccd +floatmin(Float64) / -floatmax(Float64) + +# ╔═║ a8da2081-9fad-4be6-b486-c5cb6316d152 +md"Furthermore sensible operations involving zero and infinity are defined:" + +# ╔═║ 1c2c9d4e-259f-4894-ad5b-87b176716829 +-0.0 / -Inf + +# ╔═║ c1113c4b-55aa-4f0c-a602-14279d0702f0 +md""" +## A taste of rounding error analysis + +With the floating-point number system set up, we now want to perform rounding error analysis on a single standard application, namely the computation of an **inner product** $s_n = x^T y$ between two vectors $x, y \in \mathbb{R}^n$. This is a key building block of linear algebra and underlying many operations, including matrix-vector products and orthogonalisation ... i.e. our ingredients for the Power method. + +We thus focus in detail on the evaluation of $s_n = x_1 y_1 + \cdots + x_n y_n$, where we take the evaluation order in the sum from left to right. Using the standard model of FP, we have +```math +\begin{aligned} +\hat{s}_1 &= fl(x_1 y_x) = x_1 y_1 (x + Ξ΄_1), \\[0.5em] +\hat{s}_2 &= fl(\hat{s}_1 + x_2 y_2) \\ + &= \big( \hat{s}_1 + x_2 y_2 (1 + Ξ΄_2) \big) (1 + Ξ΄_3) \\ + &= \big( x_1 y_1 (x + Ξ΄_1) + x_2 y_2 (1 + Ξ΄_2) \big) (1 + Ξ΄_3) \\ + &= x_1 y_1 (x + Ξ΄_1) (1 + Ξ΄_3) + x_2 y_2 (1 + Ξ΄_2) (1 + Ξ΄_3), \\[0.5em] +\hat{s}_3 &= fl(\hat{s}_2 + x_3 y_3) \\ + &= \big( \hat{s}_2 + x_3 y_3 (1 + Ξ΄_4) \big) (1 + Ξ΄_5) \\ + &= \big( x_1 y_1 (x + Ξ΄_1) (1 + Ξ΄_3) + + x_2 y_2 (1 + Ξ΄_2) (1 + Ξ΄_3) + + x_3 y_3 (1 + Ξ΄_4) \big) (1 + Ξ΄_5) \\ + &= x_1 y_1 (x + Ξ΄_1) (1 + Ξ΄_3) (1 + Ξ΄_5) + + x_2 y_2 (1 + Ξ΄_2) (1 + Ξ΄_3) (1 + Ξ΄_5) + + x_3 y_3 (1 + Ξ΄_4) (1 + Ξ΄_5) +\end{aligned} +``` +where $|Ξ΄_i| \leq u$. The pattern becomes clear: Assuming for simplicity $1 + Ξ΄_i = 1 + Ξ΄$ then +```math +\hat{s}_n = x_1 y_1 (x \pm Ξ΄)^n + x_2 y_2 (1 \pm Ξ΄)^n + + x_3 y_3 (1 \pm Ξ΄)^{n-1} + \cdots + x_n y_n (1 \pm Ξ΄)^2. +``` +""" + +# ╔═║ 474ef698-cb16-44a3-b7fd-5a01005278e9 +md""" +To simplify this we employ the following *(Higham, Lemma 3.1)* + +> **Lemma.** If $|Ξ΄_i| \leq u$ and $ρ_i = \pm 1$ for $1 \leq i \leq n$ and $n u < 1$, then +> ```math +> \prod_{i=1}^n (1 + Ξ΄_i)^{ρ_i} = 1 + ΞΈ_n, +> ``` +> where +> ```math +> |ΞΈ_n| \leq \underbrace{\frac{nu}{1 - nu}}_{=\gamma_n}. +> ``` + +which results in + +```math +\hat{s}_n = x_1 y_1 (1 + ΞΈ_n) + x_2 y_2 (1 + ΞΈ'_n) + + x_3 y_3 (1 + ΞΈ_{n-1}) + \cdots + x_n y_n (1 + ΞΈ_2). +``` + +This result depends on the order of summation (since a smaller backward error is associated to $x_n y_n$, which we sum last). It is easy to see, that a less tight result is (in vector notation) +```math +fl(x^T y) = (x + Ξ”x)^T y = x^T (y + Ξ”y), \qquad |Ξ”x| \leq Ξ³_n |x|, \qquad |Ξ”y| \leq Ξ³_n |y|, \qquad \text{$(\ddagger)$}, +``` +which is notably independent on the summation order. The operations $|\,\cdot\,|$ and $\leq$ are to be understood element-wise. +""" + +# ╔═║ b85874a4-d9ce-4131-9b1f-beaf79659b82 +md""" +This is the *backwards error result*, which can be interpreted as follows: The computed inner product is the exact one for a perturbed set of data $y + Ξ”y$, where each component-wise perturbation is bounded by $Ξ³_n = \frac{nu}{1 - nu}$, which is tiny (e.g. around $10^{-12}$ for `Float64` and $n = 10\,000$). Alternatively one could leave $y$ fixed and consider similar perturbations $x + Ξ”x$. + +A *forward error* can be easily obtained from $(\ddagger)$ as +```math +\left|x^T y - fl(x^T y)\right| \leq Ξ³_n \sum_{i=1}^n |x_i y_i| = Ξ³_n |x|^T |y|. +``` +""" + +# ╔═║ f02e854d-50bb-43dc-8b4d-b85df11dc739 +md""" +**Proof of the Lemma.** We consider the case $ρ_i = 1$, the case $ρ_i = -1$ proceeds similarly. Thus, we are asked to prove that if $|Ξ΄_i| \leq u$ for $1 \leq i \leq n$ and $n u < 1$, then +```math +\prod_{i=1}^n (1 + Ξ΄_i) = 1 + ΞΈ_n, +\qquad \text{where} \quad |ΞΈ_n| \leq \frac{nu}{1 - nu}. +``` +For this we proceed by induction over $n$. First we note that +```math +|Ξ΄_1| \leq u \leq \frac{u}{1 - u} \qquad \text{since $u < 1$}, +``` +therefore for $n = 1$ a valid $\theta_1$ is simply $Ξ΄_1$. Now we consider $n > 1$ and note +```math +\prod_{i=1}^n (1 + Ξ΄_i) = (1 + Ξ΄_n) (1 + ΞΈ_{n-1}) = 1 + ΞΈ_n +``` +Therefore, $ΞΈ_n = Ξ΄_n + (1 + Ξ΄_n) ΞΈ_{n-1}$ and +```math +\begin{aligned} +|ΞΈ_n| &\leq u + (1+u) \frac{(n-1) u}{1 - (n-1) u} \\ +&= \frac{u \big(1 - (n-1)u\big) + (1+u)(n-1) u}{1 - (n-1) u} \\ +&= \frac{nu}{1- (n-1)u}\\ +&\leq \frac{nu}{1- nu}. +\end{aligned} +``` +""" + +# ╔═║ ffb4717a-5059-4381-9de9-5d9e41a19183 +md""" +## Taming floating-point errors + +Having discussed a number of things that can go wrong with floating-point computations, we now want to give some techniques that aid in detecting or fixing problems. For more details see Section 4 for of *Boldo et al.*. + +### Fused multiply-add (FMA) + +One of the more recent additions to IEEE-754 is the requirement for vendors to ship a fused multiply-add (FMA) with correct IEEE-conform rounding. This operation can be defined as +```math +\text{FMA}(a, b, c) = ab + c, +``` +but where notably no intermediate rounding takes place after the multiplication of $a$ with $b$. Since this sequence of operations is a key component a number of standard primitives (dot products, multiplication of complex numbers, polynomial evaluations, ...), the ability to avoid intermediate rounding can be exploited to improve accuracy. +""" + +# ╔═║ 8f9e4905-5c39-4ccc-b0b5-af58e86df04c +md"""A standard 2x2 determinant function is:""" + +# ╔═║ 1343eaa4-d899-4ccd-8fbc-0f813e3a7ba6 +function determinant_2x2(M) + (M[1, 1] * M[2, 2]) - (M[1, 2] * M[2, 1]) +end + +# ╔═║ da63cabb-5f02-467f-9a30-87c4d4e37288 +md"An alternative by Kahan using FMAs:" + +# ╔═║ 3491e075-aac0-418d-8c01-dfee9d19bfd8 +@bind logΞ΅ PlutoUI.Slider(-16:1:0; show_value=true, default=-8) + +# ╔═║ 028b6e6f-e514-43ed-85ca-884b77000fb4 +md"""Kahan's FMA-based determinant achieves high relative accuracy throughout the range of $Ξ΅$ in `Float64` whereas the naive algorithm deteriorates for small $Ξ΅$. Similar for `Float32` the quality is overall better.""" + +# ╔═║ 30a193eb-9f22-4509-83b8-ea263884c021 +md""" +### Error-free transformations and compensated algorithms + +A perhaps surprising result, which was realised remarkably early in the design of floating-point arithmetic is *(Property 2.11 in Boldo et al.)* + +> Let $a, b \in F$ and $s = RN(a + b)$. If the FP addition of $a$ and $b$ does not overflow, then $s - (a+b) \in F$. + +This result guarantees, that for the `RoundNearest` rounding mode, the rounding error can also be represented as a floating-point number. This motivates the formulation of so-called **error-free transformations**, i.e. methods where we employ some extra floating-point operations in order to not only compute our desired result, but also the associated floating-point error. + +For example in the sum case, the above result assures us that we can represent the sum $a + b$ by two numbers $s + t$, where $s$ is the floating-point sum and $t$ is its error. An implementation is: +""" + +# ╔═║ 21e1d043-6b31-4632-8f8a-8148a5b18ab8 +function fastTwoSum(a, b) + s = a + b + t = b - (s - a) + (; s, t) +end + +# ╔═║ a4d9ccda-9088-4d52-ab4a-584654ef20e8 +fastTwoSum(0.1, 0.2) + +# ╔═║ 7683961e-fa0d-4960-80f6-c1af9d4593a7 +md"which indeed gives the floating-point error as the second term" + +# ╔═║ 12bfc098-7c8b-4d7a-b1e4-a2361beb70c3 +md"""However, this algorithm makes one assumption, namely that the exponent of $a$ (first argument) is larger than the exponent of $b$ (second argument). This condition is important, and otherwise the error can be *very far* from the error of the FP addition. Consider the example""" + +# ╔═║ 5adb95eb-a275-4d4d-9356-6fee5c387328 +fastTwoSum(1.0, 2.0^55) + +# ╔═║ 70616c6f-3f61-494c-a8fa-5de02dbe9a9b +md"where $t=0$ albeit clearly $t=1$. In the general case (without making assumptions on $a$ and $b$) we need to employ:" + +# ╔═║ 7f849f7f-2425-4ef9-8e1e-dd36a0aab126 +function twoSum(a, b) + s = a + b + v = s - a + t = (a - (s - v)) + (b - v) + return (; s, t) +end + +# ╔═║ 6cc3184b-5c44-4aa4-84c9-7647872bdcd3 +md"Now the answer from our previous example is correct:" + +# ╔═║ 4891a8a5-7941-4c07-8c20-cd2bf94ab827 +twoSum(1.0, 2.0^55) + +# ╔═║ 854efe00-c89f-40c0-b31e-42219acf5c44 +md""" +As spelled out here these algorithms might not yet seem extremely useful --- in particular since they need three (respectively six) times the number of operations compared to a standard summation. + +However, it is important to note that summations involving data of very unequal sizes are not unusual in scientific applications (e.g. finite differences, Euler-type integration schemes) and can easily lead to catastrophic cancellation (i.e. cancellation where no bit is correct). For example: +""" + +# ╔═║ e6fb6742-543b-4d90-b902-2f620f393b0a +md"We employ this naive summation function on a specially crafted vector of numbers of different size, which are constructed to sum to 1.0." + +# ╔═║ 3c0cdb90-1ff6-488a-a6b4-190bf000feb9 +function generate(N::Integer) # Generate a vector that sums to one by construction + x = randn(N) .* exp.(10 .* randn(N)) + x = [x; -x; 1.0] + x[sortperm(rand(length(x)))] +end + +# ╔═║ b7b2d7ff-b5b4-48c2-ad56-660cb813814b +md"""Clearly quite a significant error. Now, based on the idea of `fastTwoSum` one can design so-called **compensated summation** algorithms, i.e. algorithms that try to recover some of the accuracy that can be lost in summation due to catastrophic cancellation. + +A simple approach is Kahan's compensated summation method: +""" + +# ╔═║ 2d03b229-ba6e-447c-a13c-72132e5ccdc4 +md"""This method is notably better, but still far away from the exact answer. But based on Kahan's idea better methods have been proposed and are available as part of the `AccurateArithmetic` Julia package, for example: +""" + +# ╔═║ 6f507158-ef74-456b-b820-faaf6a4042e1 +md"""Both methods are spot-on! + +Moreover, the careful and optimised implementation of these algorithms in `AccurateArithmetic` employs modern CPU instructions (such as vectorisation) to effectively hide part of the overhead of the additional summation steps that need to be performed in the compensated implementations. The result is that despite achieving significantly better accuracy, these algorithms are only about a factor of two slower than Julia's builtin `sum` function for larger problem sizes: +""" + +# ╔═║ e5269069-4235-4660-aee1-af5e149ef29d +@bind logN PlutoUI.Slider(1:0.5:6; default=3.0, show_value=true) + +# ╔═║ f29adb10-1992-4e77-ad8c-61a4877e78d5 +v = generate(ceil(Int, 10^logN)); + +# ╔═║ 9e2fc100-4165-43fd-87e4-65ea2bba6368 +md""" +### Double-word arithmetic + +We saw above that compensated algorithms like `fastTwoSum` or `twoSum` produce exact summation results $r$ in the form of unevaluated sums $r_h + r_l$, where $r_h = RN(r)$ and $r_l$ is the remainder to obtain the exact answer. If the underlying floating-point type is `Float64` this makes it easily feasible to manipulate numbers with a precision beyond 100 bits! However, since only standard floating-point units are employed, the performance is generally faster than `BigFloat` or related arbitrary-precision libraries. + +This motivates the investigation of so-called **double-word arithmetic**, which is indeed based on a double-word (DW) number type that represents the unevaluated sum $x = x_h + x_l$ with $x_h = RN(x)$. + +A very sketchy Julia implementation, that supports summation can be obtained in a few lines of code: +""" + +# ╔═║ 3bfbecc2-c89c-4323-aff6-c7d7600767a6 +begin + # Composite type representing the unevaluated sum + struct DoubleWord + high::Float64 + low::Float64 + end + + # Cast from ordinary float to DoubleWord + DoubleWord(x::Float64) = DoubleWord(x, 0.0) + + # Function returning canonical zero in this type + Base.zero(DoubleWord) = DoubleWord(0.0, 0.0) + + # Addition DoubleWord & DoubleWord + function Base.:+(xw::DoubleWord, yw::DoubleWord) + (sh, sl) = twoSum(xw.high, yw.high) + (th, tl) = twoSum(xw.low, yw.low) + c = sl + th + vh, vl = fastTwoSum(sh, c) + w = tl + vl + zh, zl = fastTwoSum(vh, w) + DoubleWord(zh, zl) + end +end + +# ╔═║ c2daafe3-672c-4b9a-ad32-e4d773ecaeec +let + x = 1.0 + 1e-16 + y = f(x) +end + +# ╔═║ 9dee1557-ec64-4146-abd0-aaba2ad3afdc +f(1.0 + 3e-16) + +# ╔═║ 65fc5ab7-364b-4e77-924e-bd0f2d3ad44e +1 + eps(1.0) + +# ╔═║ 605a633d-2362-4ebe-8ca0-7db925e52a2e +compute_e(n::T) where {T} = (one(T) + one(T) / n)^n + +# ╔═║ 562abdb2-0a8a-4bc6-aa2c-0266de924ee6 +begin + p = plot(xaxis=:log, yaxis=:log, ylabel="absolute error", + xlabel="n", legend=:bottomleft, lw=2) + for T in (Float32, Float64, BigFloat) + ns = T(10).^(1:18) + errors = abs.(compute_e.(ns) .- β„―) + plot!(p, ns, errors, label=string(T), lw=2) + end + vline!([2/eps(Float32)], ls=:dash, label="inv. Float32 precision", c=1, lw=2) + vline!([2/eps(Float64)], ls=:dash, label="inv. Float64 precision", c=2, lw=2) + p +end + +# ╔═║ e8d4565a-9a74-473a-b8b7-87a88f933bcb +let + ks = (1:361) + samples = 1.606 .+ (ks .- 1) .* 2 .^ -52.0 + references = r.(BigFloat.(samples)) + + p = plot( ks, references, label="BigFloat", lw=2) + p = scatter!(p, ks, r.(samples), label="Float64") +end + +# ╔═║ ca3562da-7965-441d-bbaf-0b80c7edfa2d +let + function eval_with_precision(precision::Int) + setprecision(BigFloat, precision) do + x = 1 / BigFloat(7) + a = 10^(-BigFloat(8)) + b = 2 ^ BigFloat(24) + x + a * sin(b * x) + end + end + + reference = eval_with_precision(256) + precisions = 10:40 + evals = abs.(eval_with_precision.(precisions) .- reference) + plot(precisions, evals; yaxis=:log, mark=:x, label="", lw=2) +end + +# ╔═║ bbbf8e57-2ee0-44c6-9d84-d51c9fe5fd1e +let + Ξ² = 2.0 + t = 3 + emin = -1 + emax = 3 + + numbers = [0.0] + append!(numbers, [m * Ξ²^(e - t) for m in Ξ²^(t-1):(Ξ²^t-1) for e in emin:emax]) + sort!(numbers) + + @show numbers + + p = scatter(numbers, zero(numbers); label="Normalised numbers", mark=:x, ylims=(-1, 1), ms=6, legend=:topright) + vline!(p, [0.25, 0.5, 1, 2, 4], label="Factors of 2", c=3) + xlims!(p, (-0.01 * plot_xlim, plot_xlim)) +end + +# ╔═║ 75a6cd93-4609-4b93-a1f8-8e905cdc6602 +let + Ξ² = 2.0 + t = 3 + emin = -1 + emax = 3 + + numbers = [0.0] + append!(numbers, [m * Ξ²^(e - t) for m in Ξ²^(t-1):(Ξ²^t-1) for e in emin:emax]) + sort!(numbers) + + subnormals = [m * Ξ²^(emin - t) for m in 1:(Ξ²^(t-1)-1)] + @show subnormals + + p = scatter(numbers, zero(numbers); label="Normalised numbers", mark=:x, ylims=(-1, 1), ms=6) + scatter!(subnormals, zero(subnormals); label="Subnormal numbers", mark=:+, ms=6) + vline!(p, [0.25, 0.5, 1, 2, 4], label="Factors of 2", c=3) + xlims!(p, -0.1, 4.1) +end + +# ╔═║ 7108724e-5973-43bf-ab75-d4c5e6935cd1 +Inf + Inf + +# ╔═║ b5bc55d1-00ca-4733-8f32-08b4528ba297 +begin + Ξ΅ = 10.0^logΞ΅ + M = [Ο€ exp(1); 355/113 23225/8544+Ξ΅] +end + +# ╔═║ 39f6f711-eab3-4fa1-9061-ec7042010c63 +md""" +A classic example is the computation of determinants of 2x2 matrices. We consider the matrix $M$ with $Ξ΅ =$ $(Ξ΅): +""" + +# ╔═║ 39e10226-e527-4699-af33-ea64ccdcb066 +md"""Use the slider to change the definition of the matrix +$M =$ $(latexify_md(M)) and see what happens in single and double precision:""" + +# ╔═║ a8cb9060-9fd1-4006-8f65-d1882b6be082 +function det_relative_error(det_value::T) where {T} + detM_ref = det(BigFloat.(M)) + T( abs(det_value - detM_ref) / abs(detM_ref) ) +end + +# ╔═║ 1929e74c-e6b7-4ffe-b569-61248ba075c3 +relerror_naive_fp32 = det_relative_error( determinant_2x2(Float32.(M)) ) + +# ╔═║ 08a6f5a4-8c3f-4fa2-b40d-6575ba1597d0 +relerror_naive_fp64 = det_relative_error( determinant_2x2(M) ) + +# ╔═║ ba7ca52e-e293-4515-a327-af2be18e41b1 +function determinant_kahan(M) + w = M[1, 2] * M[2, 1] + e = fma(-M[1, 2], M[2, 1], w) + f = fma( M[1, 1], M[2, 2], -w) + e + f +end + +# ╔═║ 540686e3-df39-44f4-86ff-315d5e080ead +relerror_kahan_fp32 = det_relative_error( determinant_kahan(Float32.(M)) ) + +# ╔═║ 5162027a-b82f-4690-af0a-a6aabeed8652 +relerror_kahan_fp64 = det_relative_error( determinant_kahan(M) ) + +# ╔═║ 5435303f-2175-4ddd-80a0-7b069a7f8a8d +(BigFloat(0.1) + BigFloat(0.2)) - (0.1 + 0.2) + +# ╔═║ a4e5b19f-913c-406f-afe7-5d0fb6b3df3f +function sum_naive(x) + accu = zero(eltype(x)) + for xi in x + accu += xi + end + accu +end + +# ╔═║ 95b1ec78-033e-4f03-8adb-8045848ca123 +begin + sample = generate(100) + sum_naive(sample) +end + +# ╔═║ 73594ce6-2eb0-444d-bd66-b490d9d72d1e +sum_kbn(sample) # Improvement over sum_kahan by Kahan, Babuska and Neumaier + +# ╔═║ d2755839-8576-4749-8e2e-34f8b6cbda6b +sum_oro(sample) # Method by Ogita, Rump and Oishi + +# ╔═║ 39f5ca43-fa61-4f27-b610-016638a54eeb +function sum_kahan(x) + accu = zero(eltype(x)) + error = zero(eltype(x)) + for xi in x + temp = accu + y = xi + error + accu = temp + y + error = (temp - accu) + y + end + accu +end + +# ╔═║ e8476598-32e4-4c31-8d21-42835ceb36b6 +sum_kahan(sample) + +# ╔═║ 8240ab3a-5b57-4d40-8b04-e8a7bcf5d5cb +let + p = plot(xlabel="Runtime (ns)", ylabel="error", yaxis=:log, legend=:bottomleft) + for sumfunction in (sum, sum_kbn, sum_oro) + error = abs(1 - sumfunction(v)) .+ 1e-20 # Offset to avoid log(0.0) + + bench = @benchmark $sumfunction($v) # run benchmark + runtime = BenchmarkTools.median(bench).time + + scatter!(p, [runtime], [error]; label=string(sumfunction)) + end + p +end + +# ╔═║ bbd6e360-fca7-41b4-83c8-f6b03c5b6243 +md""" +Of note a posteriori error bounds are available for these algorithms, which moreover have been formally verified using the proof verification system Coq. + +Employing such a `DoubleWord` type directly improves our sums on our sample from the previous section --- where the `sample` vector was constructed to sum to `1`:""" + +# ╔═║ 2f6f1ee8-a85c-4246-ad27-19f31eb1ce0f +# Ordinary Float64: +sum(sample) + +# ╔═║ c300dde5-3a4c-43da-82c3-48b24a31ab04 +# DoubleWord-based sum +# (Note how this seamlessly integrated into Julia's own sum function!) +sum(DoubleWord.(sample)).high + +# ╔═║ eb7a1a03-7cf1-4eb1-a7ea-b3c4ca4ad268 +md""" +A fully feature-complete implementation of double-word arithmetic in Julia is provided by the `DoubleFloats` package. In combination with the `GenericLinearAlgebra` package, which provides pure-Julia implementations for all key linear-algebra operations (e.g. `eigen`, `qr`, `svd`, `cholesky` ...), this provides an excellent environment to run algorithms in increased precision for testing purposes. Notably the **runtime penalty** of using `DoubleFloats` is **usually a factor of 10 smaller** than using `BigFloat`. This makes `DoubleFloats` applicable for realistic calculations. In MatMat group we use it for example for testing our actual quantum-chemical materials simulations for floating-point stability. +""" + +# ╔═║ 95ed87cb-8a10-4b7e-8641-33277d87dab9 +md"""First we try the summation on our constructed vector from above:""" + +# ╔═║ f3909e32-5a43-4ac1-8ac1-ac9feb7d7b8b +sum(Double64.(generate(10000))) + +# ╔═║ 25156ed5-8612-4652-ab20-90afd8416c92 +md"Note the excellent agreement with the expected result of 1! + +Next we run the Power method to benchmark the performance. We generate a random dense matrix and starting vector: +" + +# ╔═║ 6f2c5dc3-410e-4011-be4c-bb267de6b539 +begin + Msample = diagm(abs.(randn(1000))) + v0 = randn(size(Msample, 2)) +end; + +# ╔═║ 62916272-9ab8-46b2-afce-c8a5c02c5a1c +md"First `Float64`:" + +# ╔═║ 58809041-6b74-4c91-a589-f581d054b88c +@btime power_method($Msample, $v0; verbose=false, maxiter=50, tol=1e-10) + +# ╔═║ bf73a491-7d4e-48d9-8c93-d6fbef1b7c3f +md"Then `Double64`, which is about 100 times slower:" + +# ╔═║ 01f1ef22-b7b4-4a9d-9378-98547f379902 +let + Md = Double64.(Msample) + v0d = Double64.(v0) + @btime power_method($Md, $v0d; verbose=false, maxiter=50, tol=1e-10) +end + +# ╔═║ 804d1560-83fe-4455-8277-f3ce884c4887 +md"In contrast with `BigFloat` we pay a factor of 1000:" + +# ╔═║ 7640338a-daf2-4f83-b1d7-0a3b95c86508 +let + Md = BigFloat.(Msample) + v0d = BigFloat.(v0) + @btime power_method($Md, $v0d; verbose=false, maxiter=50, tol=1e-10) +end + +# ╔═║ 35e54531-a0c6-4419-9223-423d86ced2a2 +md""" +### Interval arithmetic + +The idea of interval arithmetic is to replace computation employing numbers by **computation employing sets**. To motivate why this might be a good idea we will consider yet another great example by William Kahan. + +Consider the function +""" + +# ╔═║ 11b8bda7-a311-47bd-8974-780856ee1e54 +begin + f_interval(x) = (1/80) * log(abs(3*(1 - x) + 1)) + x^2 + 1 + + plot(0.5:0.01:2.0, f_interval, leg=false) +end + +# ╔═║ 3cbaf02a-ad01-4b45-bbdf-0f89e685bb10 +md"The function looks smooth except for a little blip around 1.3. Let's zoom in a little more closely to see what's going on:" + +# ╔═║ e15ce06b-f435-4f64-aaf2-1285faa1e4db +plot(1.2:0.0001:1.5, f_interval, leg=false) + +# ╔═║ 017d6bfc-7ad5-4f44-b30a-a3f9aec70e97 +plot(f_interval, 1.2, 1.5, leg=false) + +# ╔═║ 8a388e43-c16c-432a-839d-9ca405331ee1 +plot(f_interval, 1.32, 1.34, leg=false) + +# ╔═║ b389079a-ce96-4111-946d-e98d372e529b +md"Looks like a pretty sharp cusp. In fact if we stare at the original function for a while we see that the function diverges to $-\infty$ at $x = 4/3$. + +But $4/3$ is not representable as a binary floating-point number. So what is happening at its neighbours: +" + +# ╔═║ edf7d11c-258a-4666-9286-095a8f63fbbc +begin + x_minus = Float64(4//3, RoundDown) + f_interval(x_minus) +end + +# ╔═║ b428c6f0-5812-40fc-8b75-1cd8713162ef +begin + x_plus = nextfloat(x_minus) + f_interval(x_plus) +end + +# ╔═║ 617b7c50-9041-445e-babf-cb008790af90 +md""" +So in between the *neighbouring* floating-point numbers `x_minus` and `x_plus` the function crashes down to $-\infty$ and back. If we stick to `Float64` we can *never* see this. + +Unfortunately even changing to `BigFloat` does not change too much: +""" + +# ╔═║ 8baec096-45c7-4300-bd14-0bd8b34eceb8 +let + x_minus_big = BigFloat(4//3, RoundDown) + f_interval(x_minus_big) +end + +# ╔═║ c50598ab-0041-4d3d-9f4e-c138b68abe8b +let + x_plus_big = nextfloat(BigFloat(4//3, RoundDown)) + f_interval(x_plus_big) +end + +# ╔═║ 7d3772a8-f335-4944-950e-252403a1ca74 +md""" +One way to make sure we at least detect this behaviour is if we were able to compute $f(x)$ for *all* real $x$ in a certain *interval*, say in this case $\mathbf{x} = [1.2, 1.5]$ (we will denote intervals in bold). In other words we want to compute the range of $f$ over the input set $\mathbf{x}$, which we denote by $f(\mathbf{x})$. If $\mathbf{x}$ is a closed and bounded interval and $f$ is continuous than the range will also be a closed and bounded interval. + +The idea to be able to determine the value range of functions given an input interval has lead to the development of **interval arithmetic**. Its main contribution is to provide a cheap computational receipe to calculate an *over-approximation* of the range of $f$, i.e. some interval $\mathbf{y}$ satisfying $f(\mathbf{x}) \subset \mathbf{y}$. + +Let us first clarify on the term **interval**. In the context of interval arithmetic this always refers to a closed interval $\mathbf{x} = \{x \in \mathbb{R}\ |\ a \leq x \leq b\} \subseteq \overline{\mathbb{R}}$ where $a, b \in \overline{\mathbb{R}} = \mathbb{R} \cup \{\pm \infty\}$. + +Now instead of approximating a quantity $x$ by a single floating-point number $\hat{x}$ we can alternatively approximate it by an interval $\mathbf{x}$ spanning the next smallest and next largest representable floating-point number. For example +""" + +# ╔═║ abe3fbe7-8354-4ea7-a73f-4d0361caa58a +four_thirds = I"4//3" # Make interval representing the rational number 4/3 + +# ╔═║ dd9d43ec-d467-4af9-8910-44ef39c6bfa7 +typeof(four_thirds) + +# ╔═║ 2e9ca4cd-ade1-46b0-b57c-cb7fbee1bdcd +md"""This is a Julia interval, which has two fields `lo` and `hi`, representing the interval [`lo`, `hi`].""" + +# ╔═║ 21495bca-2cb2-458b-a7d0-3c4fd1470d1b +four_thirds.bareinterval.lo # The Float64 number smaller than 4/3 + +# ╔═║ 23d35d53-20a4-42a1-828d-823dbfa14aa2 +four_thirds.bareinterval.hi # The Float64 number larger than 4/3 + +# ╔═║ 6142c138-d5dc-4e62-8d1a-9df8f8f8c7cd +md"""On top of this we define our operations on these intervals in a way the *containment property* is guaranteed. For example the difference between two intervals $\mathbf{x}$ and $\mathbf{y}$ is computed, such that it satisfies the following property: +```math +\forall x, y \in \overline{\mathbb{R}}: \qquad x \in \mathbf{x} \text{ and } y \in \mathbf{y} \Longrightarrow x - y \in \mathbf{x} - \mathbf{y} +\qquad \text{$(\ast)$} +``` +One way to *compute* such a derivative interval $\mathbf{x} - \mathbf{y}$ is to ask : What are the smallest and largest values that $x - y$ can take, given that $x\in\mathbf{x}$ and $y\in\mathbf{y}$. More algorithmically if $\mathbf{x} = [x_\text{lo}, x_\text{hi}]$ and $\mathbf{y} = [y_\text{lo}, y_\text{hi}]$, then +```math + \mathbf{x} - \mathbf{y} = [RD(x_\text{lo} - y_\text{hi}), RU(x_\text{hi} - y_\text{lo})]. +``` +Note that this choice of rounding modes along with a proper IEEE implementation of overflow and rounding are crucial in order to be able to provably obtain the containment propery $(\ast)$ above. Packages like `IntervalArithmetic` provide such operations adhering rigorously to the containment propery for pretty much all basic operations (`*`, `+`, `/`, functions like `sqrt`, ...). The full strength of interval arithmetic now results from the + +**Fundamental theorem of interval arithmetic.** Given a function $f(x)$, which can be decomposed into a sequence of elementary operations $\left(e_1 \circ e_2 \circ \cdots \circ e_n\right)(x)$ and for each elementary operation an interval-analogue $e_i(\textbf{x})$ satisfying the containment property can be found, then by induction the end result of the calculation $\left(e_1 \circ e_2 \circ \cdots \circ e_n\right)(\textbf{x})$ is a guaranteed enclosure for the range of the original function $f$. + +**Strength of interval arithmetic.** A direct consequence of the above theorem is that if we perform our computation using intervals and the resulting interval is narrow, then this is a **mathematical proof** that the exact answer can be found within the resulting interval. We can see this as a kind of automatic run-time error analysis. + +For example the computation +""" + +# ╔═║ f64e2ffd-075f-4c68-b7c7-433736c0aea7 +sum_naive(interval.(sample)) + +# ╔═║ 9309b19e-6eed-41a8-b95e-fa04cda1cc4c +md""" +tells us that albeit the `sum_naive` function has some instabilities if the `sample` triggest too much cancellation, in this particular instance a few digits are *provably* still trustworthy. + +**Issues with interval arithmetic.** First and foremostly interval arithmetic doubles the storage cost (two floats for each number) and increases the computational cost by a factor of $4$. +But the more problematic aspect is that the computed enclosures are *usually* too wide, which can cause issues when algorithms contain branching. For example suppose one needs to check $x \leq 0$ in order to determine which code branch to follow and the enclosing interval $\mathbf{x}$ contains the $0$, i.e. the number represented by $\mathbf{x}$ might be positve *or* negative. Which path should we follow? Often enough in complicated calculations one ends up with the interval $\mathbf{x} = [-\infty, \infty]$, which of course trivially contains the exact result, but is pointless to determine its value. + +**Dependency problem.** A good example to illustrate the above issue is the dependency problem. Indeed, in many numerical computations intermediate sub-expressions can be strongly correlated, e.g. contain even common computation. Thus, their numerical error is similarly correlated --- which is completely ignored in naive interval arithmetic, which always assumes expressions to be fully independent of each other. The archetypical example is $x - x$. Irrespective of the numerical error in $x$ the result is $0$. Thus, the desired interval would just be $[0, 0]$ as well. However, when evaluating an expression $\textbf{x} - \textbf{x}$ interval arithmetic "does not know" that these are the same objects and gives a much larger interval answer, e.g. +""" + +# ╔═║ 8a1dffc9-a649-4e25-aec5-0c2e0ea8962c +let + x = interval(0.0, 1.0) + x - x +end + +# ╔═║ 22f0df08-a997-4b5c-9c52-cbcb3c87792c +md"""**Conclusion.** +The containment property of interval arithmetic **allows for a computational proof** that a floating-point **computation gives the correct answer**, and it provides a way to **compute the floating-point error**. Since intervals tend to be too wide, however, it is only a useful technique if applied to small and carefully chosen code segments. Hardly ever is it insightful to fully run a computation using intervals. Good applications for interval arithmetic in practice are therefore +- to rigorously prove expected invariances or convergence conditions, which should be satisfied if a computation has terminated successfully. For example in eigenpair computations one could use intervals solely in the computation of the residual norm $\|r\| = \|Ax - \lambda x\|$ once the iterative solver has terminated. If the interval is narrow around $0$, the problem has been provably solved exactly despite finite-precision arithmetic. +- as a red flag for numerical instability. For example if run in `Float32` the `sum_naive` returns +""" + +# ╔═║ efa2580e-ade3-411d-a9cb-116d4ae6bd91 +sum_naive(Float32.(sample)) + +# ╔═║ 434240a9-bdb4-4149-9a06-cd9312557edb +md"""which using intervals results in""" + +# ╔═║ 8f045523-9631-47b0-8d41-fe13dcbb7ca9 +sum_naive(interval.(Float32.(sample))) + +# ╔═║ 919fa773-7364-4c0e-b712-3d3cbb805509 +md"""a result that clearly should make us suspicious. Similarly, Kahan's example function to an interval containing $4/3$ warns us that the function *might* go down all the way to $-\infty$.""" + +# ╔═║ bd6676f4-60b7-414d-b9e9-e96895aa285f +f_interval(interval(1.3, 1.4)) + +# ╔═║ e75078d3-4cf9-49a0-977d-59200df1b3b2 +TableOfContents() + +# ╔═║ 079db78c-2f64-4807-bfd5-c60b44ef0a0b +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 530) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +AccurateArithmetic = "22286c92-06ac-501d-9306-4abd417d9753" +BFloat16s = "ab4f0b2a-ad5b-11e8-123f-65d77653426b" +BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +DoubleFloats = "497a8b3b-efae-58df-a0af-a86822472b78" +GenericLinearAlgebra = "14197337-ba66-59df-a3e3-ca00e7dcff7a" +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" + +[compat] +AccurateArithmetic = "~0.3.8" +BFloat16s = "~0.5.0" +BenchmarkTools = "~1.5.0" +DoubleFloats = "~1.4.0" +GenericLinearAlgebra = "~0.3.11" +HypertextLiteral = "~0.9.5" +IntervalArithmetic = "~0.22.14" +Plots = "~1.40.5" +PlutoTeachingTools = "~0.2.15" +PlutoUI = "~0.7.59" +""" + +# ╔═║ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "b8751276d2476bd549891c9a0130ec11ad180faf" + +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "6e1d2a35f2f90a4bc7c2ed98079b2ba09c35b83a" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.3.2" + 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BW-R~! literal 0 HcmV?d00001 diff --git a/06_Diagonalisation_algorithms.html b/06_Diagonalisation_algorithms.html new file mode 100644 index 0000000..21d0c8a --- /dev/null +++ b/06_Diagonalisation_algorithms.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/06_Diagonalisation_algorithms.jl b/06_Diagonalisation_algorithms.jl new file mode 100644 index 0000000..a7c31c2 --- /dev/null +++ b/06_Diagonalisation_algorithms.jl @@ -0,0 +1,3170 @@ +### A Pluto.jl notebook ### +# v0.20.0 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═║ 80ffd650-d95b-471a-9d2f-e2bc781c322e +begin + using LinearAlgebra + using LinearMaps + using PlutoUI + using PlutoTeachingTools + using Printf + using Plots + using LaTeXStrings + using HypertextLiteral +end + +# ╔═║ 2a26de48-c3b6-405b-8c07-bae036dd805d +using DFTK + +# ╔═║ e6e2ad31-0d6d-4e59-b80d-beb736759894 +md""" +# Overview of diagonalisation algorithms +""" + +# ╔═║ c4d6b47c-6ff7-4e3c-ba16-58a2e08815bc +md""" +This lecture will provide a selective overview of standard diagonalisation algorithms, contrasting their respective scope of applicability and the main ingredients. Our main angle will be to understand questions related to numerical stability in the key ingredients of the algorithms. Instead of taking a proof-guided approach, we will mostly follow a computational approach using the increased-precision and interval techniques we discussed in previous lectures. + +A more comprehensive treatment of the topic can be found in the book [Numerical Methods for Large Eigenvalue Problems](https://epubs.siam.org/doi/book/10.1137/1.9781611970739) by Youssef Saad as well as the [Lecture notes on Large Scale Eigenvalue Problems](https://people.inf.ethz.ch/arbenz/ewp/Lnotes/lsevp.pdf) by Peter Arbenz. + +In our discussion we will always take $A \in \mathbb{C}^{N\times N}$ to be a Hermitian matrix and we will seek approximations to the eigenpairs $(\lambda_i, x_i) \in \mathbb{R} \times \mathbb{C}^N$, i.e. +```math +A x_i = \lambda_i x_i +``` +where we impose an ordering $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_N$. +""" + +# ╔═║ 5729f2a4-7557-4e5c-b230-31faf6286ff3 +md""" +## Single vector iteration + +### Power method + +We already discussed the **power method** in previous lectures and exercises as a way to obtain the eigenvector $x_J$ corresponding to the largest-absolute eigenvalue. Via the Rayleigh quotient the corresponding eigenvalue $\lambda_J$ then becomes accessible as well. + +**Remark**. Keep in mind that the **largest-absolute eigenvalue** $\lambda_J$ is in general **not the same as the largest eigenvalue** $\lambda_N$, in case a negative eigenvalue is dominating. + +For completeness, our algorithmic recipe to obtain the eigenvector to tolerance $\varepsilon$ was: + +> 0. Start from a vector $x^{(0)}$, which has non-zero overlap with $x_n$. +> 1. Compute the matrix-vector product: $\tilde{x}^{(i)} \leftarrow A x^{(i)}$ +> 2. Normalise: $x^{(i+1)} \leftarrow \tilde{x}^{(i)} / \| \tilde{x}^{(i)} \|$ +> 3. If $\| x^{(i+1)} - x^{(i)} \| < \varepsilon$, exit the procedure, else $i \leftarrow i+1$ and return to 1. + +Below we provide a slightly extended version, which additionally employs the Rayleigh quotient to estimate the eigenvalue and the residual +```math +r^{(i)} = A x^{(i)} - \lambda^{(i)} x^{(i)} +``` +to check convergence. +""" + +# ╔═║ f8827f52-cba1-433f-9c14-9a0b621b2c35 +function power_method(A; x=randn(eltype(A), size(A, 2)), tol=1e-6, maxiter=100, verbose=true) + T = real(eltype(A)) + + eigenvalues = T[] + residual_norms = T[] + Ξ» = NaN + for i in 1:maxiter + normalize!(x) + Ax = A * x + Ξ» = x'Ax + norm_r = norm(Ax - Ξ» * x) + + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ» norm_r + push!(eigenvalues, Ξ») + push!(residual_norms, norm_r) + norm_r < tol && break + + x = Ax + end + converged = residual_norms[end] < tol + !converged && verbose && @warn "Power method not converged." + + (; Ξ», x, converged, eigenvalues, residual_norms) +end + +# ╔═║ eb66e66c-278f-465e-a56d-cac7432723f7 +@bind logΞ΄ PlutoUI.Slider(-1:0.1:2; show_value=true, default=1) + +# ╔═║ 23f9fd84-0b58-4ade-a72c-b3e303a79480 +Apower = Diagonal([1, 30 - 10^logΞ΄, 30 + 10^logΞ΄]) + +# ╔═║ bfff1b7f-bf26-4373-a499-014b871067ad +let + (; residual_norms) = power_method(Apower, verbose=false, tol=1e-6, maxiter=200) + plot(residual_norms, yaxis=:log, ylims=[1e-6, 1e2], label="residual_norm", xlims=[0, 200], lw=2) + + Ξ»s = sort(eigvals(Apower)) + plot!(1:200, abs(Ξ»s[end-1] / Ξ»s[end]).^(1:200); label="Expected rate", ls=:dash, c=1, lw=2) +end + +# ╔═║ 72264fdd-d151-41b3-abd7-3fd6b0b4fb84 +md""" +From playing with the slider we notice that when the two **largest-absolute eigenvalues get too close**, the **convergence deteriorates**. This can be understood using the following theorem: + +**Theorem 5.1** Assume that $|\lambda_J| \neq |\lambda_{J-1}|$, i.e. that the largest eigenvalue is simple. Then the initial vector $x^{(0)}$ has either no overlap with the eigenvector $x_J$ or the sequence $x^{(t)}$ generated by the power method converges to $x_J$. + +This is a well-known result for which proofs can be found e.g. in the book by Saad or [on wikipedia](https://en.wikipedia.org/wiki/Power_iteration). The key outcome of the proof is that the convergence rate depends on the ratio +```math +\frac{|Ξ»_{J-1}|}{|Ξ»_{J}|} +``` +thus deteriorates as $|Ξ»_{J-1}|$ and $|Ξ»_{J}|$ are too close, e.g. if $Ξ»_J$ is a multiple eigenvalue. +""" + +# ╔═║ fa98f7d8-bc72-4ac8-b46a-2456e37e42de +md""" +### Spectral transformations + +A natural question to ask now how this method can be extended, e.g. to compute the smallest eigenvalue or eigenvalues in the middle of the spectrum of a matrix. The answer to this provide the following well-known **spectral transformations**: + +**Theorem 5.2** (Spectral transformations)**.** Assume $(Ξ»_i, x_i)$ is an eigenpair of $A$, then +1. If $A$ is invertible, then $(\frac{1}{Ξ»_i}, x_i)$ is an eigenpair of $A^{-1}$. +2. For every $Οƒ \in \mathbb{R}$ $(Ξ»_i - Οƒ, x_i)$ is an eigenpair of $A - Οƒ I$. +3. If $A - Οƒ I$ is invertible, then $(\frac{1}{Ξ»_i - Οƒ}, x_i)$ is an eigenpair of $(A - ΟƒI)^{-1}$. + +**Proof:** 1. and 2. follow from a simple direct calculation. 3. is a corollary of 1 & 2. +""" + +# ╔═║ 291cb850-5089-480e-b803-1e2a20852b95 +md""" +Part 1 of this theorem is the justification for using the **inverse power method** to approximate the smallest eigenpair of $A$ as done already in one of the previous exercises. Part 3 of this theorem, called **shift and invert** provides us with a recipe to compute an arbitrary isolated eigenvalue. + +To illustrate this we consider the matrix +""" + +# ╔═║ 98c91264-19cc-41e5-951a-a201601fbfc5 +Ashift = [0.4 -0.6 0.2; -0.3 0.7 -0.4; -0.1 -0.4 0.5] + +# ╔═║ aea3267b-1164-418c-98a8-70cc37d1c959 +md"""which has eigenvalues""" + +# ╔═║ e33b4164-b980-446f-9398-9ec68cf224f9 +eigvals(Ashift) + +# ╔═║ 7661b0fa-839a-47a1-a0ea-a89e8d4abc3a +md"""Suppose we **don't know the exact eigenvalues**, but we know that the eigenpair $(Ξ»_i, x_i)$ we are interested in has an **eigenvalue around $0.4$**: +- Then we set $Οƒ = 0.4$ and employ the matrix $(A - Οƒ I)^{-1}$ in the `power_method`, which as per our previous discussion will converge to the largest absolute eigenvalue of $(A - Οƒ I)^{-1}$. +- The eigenvalues of the latter matrix are $\left\{\frac{1}{Ξ»_i - Οƒ} \, \middle| \, i =1,\ldots,3 \right\}$. Clearly, the largest-absolute one of these eigenvalues is attained for the $Ξ»_i$ closest to $Οƒ$. +- From Theorem 5.2 we additionally know that the eigenvector of $(A - Οƒ I)^{-1}$ associated to $\frac{1}{Ξ»_i - Οƒ}$ is exactly the eigenvector of $A$ associated to $Ξ»_i$. Therefore, we have found a way to compute the *exact* eigenpair $(Ξ»_i, x_i)$ based on the approximate eigenvalue $Οƒ = 0.4$. + +Computationally we do this as such: +""" + +# ╔═║ f923e4ba-1833-4b2e-968d-8527fc70720d +let + Ξ»β‚‚ = eigvals(Ashift)[2] # Exact targeted eigenvalue + + Οƒ = 0.4 # Our approximation to the eigenvalue of interest + shifted = Ashift - Οƒ * I # Shift the matrix + factorised = factorize(shifted) # Factorise to obtain fast \ operation + + # Use a LinearMap to represent the inverse lazily + # (i.e. `inverted * v = factorised \ v`) + shift_and_invert = InverseMap(factorised) + + # Run power method on (A - ΟƒI)⁻¹ + (; Ξ», x) = power_method(shift_and_invert) + + # Compute eigenvalue using the Rayleigh quotient of the original matrix + Ξ»_rayleigh = dot(x, Ashift, x) + + println() + println("Power method converged to eigenvalue: $Ξ»") + println("Exact value for 1/(Ξ»β‚‚ - Οƒ) is: $(1/(Ξ»β‚‚ - Οƒ))") + println("Computed eigenvalue of Ashift: $Ξ»_rayleigh") + println("Exact eigenvalue using eigvals: $Ξ»β‚‚") +end + +# ╔═║ 6a049358-0e6c-4ea9-8950-52f5bdda287e +md""" +## Rayleigh quotient iterations + +We already noted earlier that the Rayleigh quotient is a natural approximation for the eigenvalue corresponding most closely to an approximate eigenvector $v$. A natural idea is thus to replace the shift $Οƒ$ in the shift-and-invert strategy by the Rayleigh quotient $R_A(x^{(i)})$ of the current iterate. + +This is theoretically backed up by the following result. + +**Proposition 5.3.** Let $A \in \mathbb{C}^{N\times N}$ be a Hermitian matrix and $v\in \mathbb{C}^N$. Then +```math +R_A(v) = \frac{v^H A v}{v^H v} = \text{argmin}_{Ξ± \in \mathbb{C}} \ \| A v - Ξ± v \|. +``` + +**Proof:** We denote $ρ = R_A(v)$ and consider the function +```math +f(z) = \|A v - (ρ + z) v\|^2 +``` +where $z \in \mathbb{C}$. We observe that +```math +\begin{aligned} +f(z) &= v^H A^2 v - (2ρ + z + \overline{z})\, v^H A v + |ρ + z|^2 v^H v \\ +&= v^H A^2 v - 2\big(ρ + \text{Re}(z)\big) \, v^H A v + \big( ρ^2 + 2ρ \text{Re}(z) + |z|^2 \big) v^H v\\ +&= v^H A^2 v + \big(-2ρ^2 - 2ρ\text{Re}(z)\big)v^H v + \big( ρ^2 + 2ρ \text{Re}(z) + |z|^2 \big) v^H v\\ +&= v^H A^2 v + (- ρ^2 + |z|^2) v^H v +\end{aligned} +``` +Since $f(z) \geq 0$ by construction and $v^H v \geq 0$, +this function is minimal for $z = 0$, +which concludes the proof. + +With this setup we obtain **Rayleigh-quotient iteration (RQI)**: +""" + +# ╔═║ b0e6abc4-7581-4a7b-9160-792d2824a1cf +function rayleigh_quotient_iteration(A; x=randn(eltype(A), size(A, 2)), + tol=1e-6, maxiter=100, verbose=true) + T = real(eltype(A)) + + eigenvalues = T[] + residual_norms = T[] + Ξ» = NaN + for i in 1:maxiter + normalize!(x) + Ax = A * x + Ξ» = x'Ax + norm_r = norm(Ax - Ξ» * x) + + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ» norm_r + push!(eigenvalues, Ξ») + push!(residual_norms, norm_r) + norm_r < tol && break + + # Note: In the Power method this was u = Au + x = (A - Ξ»*I) \ x + end + converged = residual_norms[end] < tol + !converged && verbose && @warn "RQI not converged." + + (; Ξ», x, converged, eigenvalues, residual_norms) +end + +# ╔═║ 6be8652b-525c-4efa-bcc7-b72285ea2d98 +md"""RQI is a very fast algorithm, e.g. notice""" + +# ╔═║ ab7f07f6-20f3-465f-ae46-e694100541b8 +Arqi = Diagonal(randn(100)); + +# ╔═║ 1f9b91e3-28f4-4e06-8a3d-70c84432007b +x0_rqi = randn(100); + +# ╔═║ 0f7709e7-fec9-4176-bf58-96f2d891e8b4 +md"""Notice, how the convergence in the last few steps is *very fast*, meaning that multiple orders of magnitude in the residual norm are gained in each step. + +The RQI has fantastic rates of convergence. However, it has two disadvantages: +- The main issue is that *at each RQI iteration* the matrix `A - Ξ»*I` is different and needs to be **freshly factorised** in order to compute the `\` operation. In other words we pay a full Gaussian elimination each RQI iteration. This in contrast to the *shift-and-invert* power method, where factorisation was computed once and for all for `A - Οƒ*I` *before* the `power_method` was called. +- The second point to note is that it is **hard to control to *which* eigenpair** the RQI converges. Try re-running the above a few times by updating the initial guess `u0_rqi`. +- Moreover, it is possible for the RQI to *not* converge --- albeit the set of possible initial guesses that leads to a non-converging RQI is of measure zero. It is thus most useful if one already has a decent guess for the eigenvector. + +These observations are summarised in the following +""" + +# ╔═║ 73e61d6c-0762-41d9-a0c1-c2a64926a40a +md""" +**Theorem 5.4.** For all starting vectors `u`, except a set of measure zero, the Rayleigh quotient iteration converges to an eigenpair. When it converges, the convergence rate is cubic: If $Ξ»_k$ is an eigenvalue to $A$ and `u` sufficiently close to the eigenvector $x_k$ corresponding to $Ξ»_k$, then as $i \to \infty$ +```math +|Ξ»^{(i+1)}_k - Ξ»_k| = O\big(|Ξ»^{(i)}_k - Ξ»_k|^3\big) +``` +and +```math +|x^{(i+1)}_k - x_k| = O\big(|x^{(i)}_k - x_k|^3\big), +``` +where $(Ξ»^{(i)}_k, x^{(i)}_k)$ is the approximation to $(Ξ»^{(i)}_k, x^{(i)}_k)$ the algorithm produces in iteration $i$. Here we choose $x_k$ among $\pm x_k$, such that $\langle x^{(i)}_k, x_k \rangle > 0$. + +""" + +# ╔═║ b48af18e-c65b-4ddd-b6d1-6b25e4e4bb80 +md""" +To illustrate the point about the **target of RQI's convergence being hard to predict** consider the following example. We initialise the RQI using a random starting vector +""" + +# ╔═║ b8bb7d0b-21c4-4746-9dc7-dc47f76a0736 +xstart = randn(eltype(Arqi), size(Arqi, 2)) + +# ╔═║ 009c780c-d1cc-4421-86c7-74e5a99646f9 +md""" +and allow ourself to select an *initial* shift $Οƒ$, +which we store in the variable `ΟƒD`: +- `ΟƒD = `$(@bind ΟƒD Slider(-0.2:0.01:0.2; default=0.01, show_value=true)) + +Then we define a version of `rayleigh_quotient_iteration`, which starts the procedure based on applying the matrix $(A - Οƒ I)^{-1}$ to this random vector: +""" + +# ╔═║ 4e208b64-6b88-4430-9bc3-7198c54b00ff +function rayleigh_quotient_iteration(A, Οƒ::Number, x; kwargs...) + x = (A - Οƒ*I) \ x + rayleigh_quotient_iteration(A; x, kwargs...) +end + +# ╔═║ ce876450-eea7-4088-9aca-ed3ee462cb09 +rayleigh_quotient_iteration(Arqi, x=x0_rqi) + +# ╔═║ 8b2f116a-09e6-4bbf-9a83-b7c38fe34680 +let + # exact eigenvalues of Arqi + Ξ»_rqi = eigvals(Arqi) + + # The eigenvalue the iteration targets + i = argmin(abs.(Ξ»_rqi .- ΟƒD)) + Ξ»target = Ξ»_rqi[i] + + p = plot(title="Convergence", + ylabel="residual", xlabel=L"i", legend=:bottomleft, + yaxis=:log, ylims=(1e-10, 3), xlims=(1, 7)) + + q = plot(title="History", xlabel=L"i", ylabel=L"Ξ»_k^{(i)}", + legend=:bottomleft, ylims=(-0.2, 0.2), xlims=(1, 7)) + hline!(q, Ξ»_rqi, ls=:dash, label=L"eigenvalues of $A$", c=2, lw=1.5) + hline!(q, [ΟƒD], ls=:dash, label=L"initial shift $Οƒ$", c=3, lw=1.5) + + if abs(ΟƒD - Ξ»target) > 1e-10 # Guard to prevent numerical issues + results = rayleigh_quotient_iteration(Arqi, ΟƒD, xstart; maxiter=7) + plot!(p, results.residual_norms; mark=:o, label="", lw=1.5) + plot!(q, results.eigenvalues, mark=:o, c=1, label="", lw=1.5) + end + plot(p, q; layout=(1, 2)) +end + +# ╔═║ c3c91935-60f7-4c4a-b43c-973566d58367 +md""" +## Subspace methods + +So far we restricted ourselves to the rather basic goal of solving only for a **single eigenvector** --- essentially by iterating only a single vector. We already saw that some eigenproblems are challenging to solve in this simple setting, namely exactly if the **largest-absolute eigenvalue has only a small or even a zero gap**. + +A natural generalisation to the power-method scheme is to iterate not one, but **multiple vectors**. Doing this naively is a little flawed as then simply each vector will converge to the eigenvector(s) with the largest-abs eigenvalue. Instead, what is needed is to ensure *in each step* that the obtained eigenvectors are orthogonal to each other. + +One way to achieve this is QR-factorisation, i.e. the factorisation $A = Q R$ into an orthogonal matrix $Q$ and an upper triangular matrix $R$. The orthogonalisation thus consists of simply keeping the $Q$ matrix: +""" + +# ╔═║ 41265945-67e6-4e34-8604-20faa6f475b7 +ortho_qr(A) = Matrix(qr(A).Q) + +# ╔═║ 8937b497-99f0-4d8f-998a-e2aad0d5aeb9 +md"""Built into a power iteration procedure and employing multiple vectors, this gives rise to the **subspace iteration** method (in this example iterating on $k=2$ vectors by default):""" + +# ╔═║ f64ab474-86f6-4048-b013-3c8969d2e125 +function subspace_iteration(A; X=randn(eltype(A), size(A, 2), 2), ortho=ortho_qr, + tol=1e-6, maxiter=100, verbose=true) + T = real(eltype(A)) + + eigenvalues = Vector{T}[] + residual_norms = Vector{T}[] + Ξ» = T[] + for i in 1:maxiter + # Full orthogonalisation + X = ortho(X) + + # Just normalise ... for testing + # for c in eachcol(X) + # normalize!(c) + # end + + AX = A * X + Ξ» = diag(X'*AX) # Rayleigh quotient evaluation in all vectors + push!(eigenvalues, Ξ») + + residuals = A * X - X * Diagonal(Ξ») # compute all residuals + norm_r = norm.(eachcol(residuals)) # compute all residual norms + push!(residual_norms, norm_r) + + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ»[end] norm_r[end] + maximum(norm_r) < tol && break + + X = AX + end + + (; Ξ», X, eigenvalues, residual_norms) +end + +# ╔═║ d2ef7881-bd1b-4bd3-aee4-025c7da5ba71 +@bind logsubΞ΄ PlutoUI.Slider(-1:0.1:1; show_value=true, default=1) + +# ╔═║ d21b8a45-7129-4038-82ac-250436ff8583 +Asubspace = Diagonal([150., 100. + 10^logsubΞ΄, 100., 50., 40., 30.]) + +# ╔═║ 9d5e0e5d-9117-4308-b133-20e3d9e0894e +md"The method yields the following eigenvalues:" + +# ╔═║ bd576cd6-ee70-43b3-ad05-9307d6eeb05d +subspace_iteration(Asubspace; X=randn(size(Asubspace, 2), 2), verbose=false, tol=1e-6, maxiter=200).Ξ» + +# ╔═║ 6f81e36d-7892-42ec-aeb6-91fca5b6399a +let + (; residual_norms) = subspace_iteration(Asubspace; X=randn(size(Asubspace, 2), 2), + verbose=false, tol=1e-6, maxiter=200) + norms_first = [n[1] for n in residual_norms] + norms_second = [n[2] for n in residual_norms] + + p = plot(; yaxis=:log, ylims=[1e-6, 1e2], xlims=[0, 200]) + plot!(p, norms_first, label="residual first eigenpair", c=1, lw=2) + plot!(p, norms_second, label="residual second eigenpair", c=2, lw=2) + + Ξ»s = sort(eigvals(Asubspace)) + plot!(p, 1:200, abs(Ξ»s[end-1] / Ξ»s[end]).^(1:200); + label="Rate first", ls=:dash, c=1, lw=2) + plot!(p, 1:200, abs(Ξ»s[end-2] / Ξ»s[end-1]).^(1:200); + label="Rate second", ls=:dash, c=2, lw=2) +end + +# ╔═║ 137f3b99-d031-4df4-868e-9b5e4d342a83 +md""" +We see that each eigenpair $(Ξ»^{(i)}_k, x^{(i)}_k)$ converges with its own speed, which depends on the ratio +```math +\frac{|Ξ»_{k+1}|}{|Ξ»_{k}|}, +``` +which can also be proven rigorously. +""" + +# ╔═║ 357cd9c6-eb57-48ca-afe5-a752addc6461 +md""" +## Projection methods + +The above `subspace_iteration` addresses the question of how to compute multiple eigenpairs at once. However, compared to Rayleigh quotient iteration it feels a little like a step back since we are no longer making use of the Rayleigh quotient. Notably, when computing the approximate eigenpairs we are considering each iterated vector separately and allow no "interaction" between them, even though the first iterated vector might well contain a contribution to the eigenvector, which the second iterated vector attempts to approximate. Moreover, we are right now tied to using exactly one iterated vector per eigenvector we want to approximate. We will now see how projection methods allow us to overcome both these restrictions. + +The general idea is to somehow construct an $m$-dimensional subspace $\mathcal{S} \subset \mathbb{C}^N$ with $m \ll N$ (details will be discussed later) and fully **solve the eigenvalue problem *projected* into this subspace**. We consider the basic eigenproblem for $A \in \mathbb{C}^{N \times N}$ Hermitian: +```math +A x_i = Ξ»_i x_i \qquad \text{$(\ddagger)$} +``` +An orthogonal projection technique now seeks approximate eigenpairs $(\tilde{\lambda}_i, \tilde{x}_i)$ with $\tilde{\lambda}_i \in \mathbb{R}$, $\tilde{x}_i \in \mathcal{S}$, such that the following **Galerkin condition** +```math +\big(A \tilde{x}_i - \tilde{Ξ»}_i \tilde{x}_i \big) \perp \mathcal{S} \qquad \text{(G)} +``` +is satisfied. Equivalently one may enforce +```math +\left\langle v, A \tilde{x}_i - \tilde{Ξ»}_i \tilde{x}_i \right\rangle = 0 \qquad \forall v \in \mathcal{S}. \qquad \text{$(\ast)$} +``` +Suppose now that $\{v_1, v_2, \ldots, v_m\}$ is an orthonormal basis for $\mathcal{S}$ and denote by $V = (v_1, v_2, \ldots, v_m) \in \mathbb{C}^{N\times m}$ the matrix with these basis vectors as columns. Expanding $\tilde{x}_i$ in this basis, i.e. introducing a vector $\tilde{y}_i \in \mathbb{C}^m$ with +```math +\tilde{x}_i = V \tilde{y}_i +``` +equation $(\ast)$ becomes +```math +\left\langle v_j, A V \tilde{y}_i - \tilde{Ξ»}_i V \tilde{y}_i \right\rangle = 0 \qquad \forall j = 1, \ldots, n. +``` +or (since $V^H V = I \in \mathbb{C}^{n \times n}$): +```math +\left(V^H A V\right) \, \tilde{y}_i = \tilde{Ξ»}_i \tilde{y}_i \qquad \text{$(\S)$} +``` +Note that this is an $m$-dimensional eigenproblem, which yields an approximate eigenpair $(\tilde{Ξ»}_i, \tilde{y}_i)$ to the original $N$-dimensional eigenpair $(\ddagger)$ above. Thus while $(\ddagger)$ might be too large to be solved by a dense linear algebra routine like `eigen`, the Galerkin approach has allowed us to reduce this to an $m$-dimensional problem, which is easier to treat with `eigen`. Algorithmically this technique is known as the **Rayleigh-Ritz procedure**, which can be summarised as follows: + +> 1. Find an orthonormal basis $\{v_i\}_{i=1,\ldots,m}$, spanning a subspace $\mathcal{S}$. +> 2. Compute $A_V = V^H A V$, with $V$ containing the basis vectors of $\mathcal{S}$ as columns. +> 3. Use `eigen` to compute $m$ eigenpairs $(\tilde{Ξ»}_i, \tilde{y}_i)$ of $A_V$. +> 4. Compute the approximate eigenvectors $\tilde{x}_i = V \tilde{y}_i$ of $A$, which yields the approximate eigenpairs of $A$ as $(\tilde{Ξ»}_i, \tilde{x}_i)$. + +Note, that the low-dimensional pairs $(\tilde{Ξ»}_i, \tilde{y}_i)$ are exact eigenpairs of the subproblem matrix $A_V$. The corresponding full-dimensional pairs $(\tilde{Ξ»}_i, \tilde{x}_i)$ are usually called **Ritz pair** with $\tilde{Ξ»}_i$ called **Ritz values** and $\tilde{x}_i$ a **Ritz vector**. + +One way to put this into practice is to employ the ideas of the `subspace_iteration` discussed above. Recall that this method generates in each iteration a set of orthogonal vectors `U` as guesses for the eigenpairs to be determined. These vectors span a subspace, which can be employed in a Rayleigh-Ritz procedure, leading to the projected subspace iterations: +""" + +# ╔═║ 7237c2e9-c0f3-4cd8-ad1f-6d5cbff35354 +function projected_subspace_iteration(A, V; + tol=1e-6, maxiter=100, verbose=true, + ortho=ortho_qr) + T = real(eltype(A)) + + eigenvalues = Vector{T}[] + residual_norms = Vector{T}[] + Ξ» = T[] + Y = nothing + for i in 1:maxiter + V = ortho(V) + + AV = A * V + Ξ», Y = eigen(Hermitian(V' * AV)) # Notice the change to subspace_iteration + # This is the Rayleigh-Ritz step + push!(eigenvalues, Ξ») + + residuals = AV * Y - V * Y * Diagonal(Ξ») + norm_r = norm.(eachcol(residuals)) + push!(residual_norms, norm_r) + + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ»[end] norm_r[end] + maximum(norm_r) < tol && break + + V = AV + end + X = V * Y + + (; Ξ», X, eigenvalues, residual_norms) +end + +# ╔═║ a9fdeb7d-e4f8-4320-8d32-792bf405a7aa +# Version to automatically select initiol guess +function projected_subspace_iteration(A; + X=randn(eltype(A), size(A, 2), 2), kwargs...) + projected_subspace_iteration(A, X; kwargs...) +end + +# ╔═║ 2e96d48d-15d9-40cc-8c1d-d8d03e755c2a +Asub = Diagonal(1.0:2.0:100.0) + +# ╔═║ e6a62b4f-9919-43b9-ae74-3cb3c1110829 +let + (; residual_norms, Ξ») = projected_subspace_iteration(Asub) + @show Ξ» + + norms_1st = [r[1] for r in residual_norms] + norms_2nd = [r[2] for r in residual_norms] + + plot( norms_1st; yaxis=:log, label="residual λ₄₉", lw=2) + plot!(norms_2nd; yaxis=:log, label="residual Ξ»β‚…β‚€", lw=2) +end + +# ╔═║ bef6d532-6131-4db7-8b7c-582895a035b1 +md""" +An **alternative notation** for subspace methods, that is sometimes handy, is given **in terms of projections**. +Making use of our basis for $\mathcal{S}$ it is easy to verify that +```math +P_\mathcal{S} = V V^H \in \mathbb{C}^{N\times N} +``` +is indeed a projection onto the subspace $\mathcal{S}$, since the idempotency relation +```math +P_\mathcal{S}^2 = \left( V V^H \right)^2 = V (V^H V) V^H = V I V^H = P_\mathcal{S} +``` +is satisfied. Noting further that $\tilde{x}_i = V \tilde{y}_i$ implies $V^H \tilde{x}_i = \tilde{y}_i$ and thus $P_\mathcal{S} \tilde{x}_i = \tilde{x}_i$, +we can rewrite $(\S)$ as +```math +P_\mathcal{S} A \tilde{x}_i = V \left(V^H A V\right) \tilde{y}_i +\stackrel{(\S)}{=} V \tilde{Ξ»}_i \tilde{y}_i += P_\mathcal{S} \tilde{Ξ»}_i \tilde{x}_i += \tilde{Ξ»}_i \tilde{x}_i, +``` +which in turn implies +```math +P_\mathcal{S} \left( A \tilde{x}_i - \tilde{Ξ»}_i \tilde{x}_i \right) = 0 +``` +--- a projection version of the Galerkin conditions (G) above. +Yet alternatively we can state +```math +P_\mathcal{S} A P_\mathcal{S} \, \tilde{x}_i += \tilde{Ξ»}_i P_\mathcal{S} \tilde{x}_i += \tilde{Ξ»}_i \tilde{x}_i, +``` +which is an eigenproblem in the projected operator $P_\mathcal{S} A P_\mathcal{S}$. +""" + +# ╔═║ 3cea29a7-fd87-4491-81a6-c53f79387e1f +md""" +The **appeal of projection methods** for computing the eigenpairs of Hermitian matrices results from the connection to the **Courant-Fisher min-max principle**, which guarantees that the eigenvalues computed by the Rayleigh-Ritz procedure satisfy a strong optimality condition. + +First we note that +```math +\langle \tilde{x}, A \tilde{x} \rangle += \big\langle \tilde{x}, \left(P_\mathcal{S} A P_\mathcal{S} \right)\, \tilde{x} \big\rangle +\qquad \forall \tilde{x} \in \mathcal{S}, +``` +from which it follows that +if $\tilde{x}$ is inside the subspace employed by an orthogonal projection approach (like the Rayleigh-Ritz procedure), the Rayleigh quotient of the projected matrix $P_\mathcal{S} A P_\mathcal{S}$ and of the original matrix $A$ are identical: +```math +\tilde{Ξ»}_1 = \min_{\tilde{x}\in \mathcal{S}, \tilde{x}\neq 0} \frac{\big\langle \tilde{x}, \left(P_\mathcal{S} A P_\mathcal{S} \right) \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle} += \min_{\tilde{x}\in \mathcal{S}, x\neq 0} +\frac{\big\langle P_\mathcal{S} \tilde{x}, A P_\mathcal{S} \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle} += \min_{\tilde{x}\in \mathcal{S}, \tilde{x}\neq 0} +\frac{\big\langle \tilde{x}, A \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle}. +``` +More generally, based on applying Courant-Fisher to the matrix $P_\mathcal{S} A P_\mathcal{S}$, we obtain the following result + +**Proposition 5.5.** The $i$-th approximate eigenvalue (counted from small to large including multiplicities) of a Hermitian matrix $A$ obtained from an orthogonal projection method to a subspace $\mathcal{S}$ satisfies +```math +\tilde{Ξ»}_i = \min_{s \subset \mathcal{S},\,\text{dim}(s) = i} \ \max_{\tilde{x}\in s,\,\tilde{x}\neq 0} \frac{\big\langle \tilde{x}, A \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle} +``` + +From this we obtain immediately the + +**Corollary 5.6.** For $i = 1, 2, \ldots, m$ the inequality +```math +Ξ»_i \leq \tilde{Ξ»}_i +``` +holds, i.e. eigenvalues are **approximated from above** in orthogonal projection methods. + +**Proof.** +```math +\tilde{Ξ»}_i = \min_{s \subset \textcolor{red}{\mathcal{S}},\,\text{dim}(s) = i} \ \max_{\tilde{x}\in s,\,\tilde{x}\neq 0} \frac{\big\langle \tilde{x}, A \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle} +\geq +\min_{s \subset \textcolor{red}{\mathbb{C}^N},\,\text{dim}(s) = i} \ \max_{x\in s,\,x\neq 0} \frac{\big\langle x, A x \big\rangle}{\langle x, x \rangle} += Ξ»_i. +``` +This is the **desired optimality**: The eigenvalue approximation found by a projection method is the **best eigenvalue approximation we can possibly find** within the chosen subspace, and it provides a rigorous upper bound to the exact one. + +Finally, the following characterisation provides a connection between the approximate eigenvalue $\tilde{Ξ»}_i$ and the corresponding approximate eigenvector $\tilde{x}_i$ obtained by an orthogonal projection approach: +```math +\tilde{Ξ»}_i += \frac{\big\langle \tilde{x}_i, A \tilde{x}_i \big\rangle}{\langle \tilde{x}_i, \tilde{x}_i \rangle} += \min_{0\neq \tilde{x} \in \mathcal{S},\,\tilde{x} \perp \tilde{\mathcal{X}}_{k-1}} \frac{\big\langle \tilde{x}, A \tilde{x} \big\rangle}{\langle \tilde{x}, \tilde{x} \rangle} +``` +where $\tilde{\mathcal{X}}_k = \text{span}(\tilde{x}_1, \ldots, \tilde{x}_k)$, +the space spanned by the $k$ first *approximate* eigenvectors. +""" + +# ╔═║ e1fd4f96-a465-46d8-8c78-cde7c5325e6f +md""" +### Forming a good subspace + +Projection methods and the Rayleigh-Ritz procedure are a key ingredient of pretty much all iterative diagonalisation approaches employed nowadays. A full discussion of the standard techniques employed to build the reduced subspace $\mathcal{S}$ is out of the scope of this lecture. An incomplete list of techniques worth mentioning are: +- If a Krylov basis is constructed and employed for $\mathcal{S}$ one obtains diagonalisation methods such as *Lanczos* or *Arnoldi*. +- Employing an orthogonal correction based on the current residual to expand and enrich the subspace leads to the *Davidson* family of methods. +- If the subspace is constructed following the idea of minimising the Rayleigh quotient, we obtain minimisation methods (inverse iterations, PCG, LOPCG). These will be discussed in the following. +""" + +# ╔═║ c56ac72f-643a-4b29-97cb-4f61426d11d5 +md""" +## Minimisation methods + +In our discussion so far we already employed the variational approach encoded in Courant-Fisher for obtaining eigenvalue problems. E.g. in order to obtain the smallest eigenvalue $Ξ»_1$ we may just +```math +Ξ»_1 = \min_{0 \neq x \in \mathbb{C}^N} R_A(x) \qquad R_A(x) = \frac{x^H A x}{x^H x} +``` +The purpose of this subsection is to develop methods, that employ this minimisation property directly for computing the eigenpairs of $A$. + +The natural reflex is to employ a steepest descent approach. +We thus compute the gradient of $R_A(x)$, namely +(denoting by $e_i$ the unit vector along coordinate direction $i$): +```math +\begin{aligned} +\frac{\partial R_A(x)}{\partial x_i} +&= \left. \frac{d R_A(x + Ξ΅ e_i)}{d Ξ΅} \right|_{Ξ΅=0} \\ +&= \frac{d}{d Ξ΅} \left. \left( + \frac{(x + Ξ΅e_i)^H A (x + Ξ΅e_i)}{(x + Ξ΅e_i)^H (x + Ξ΅e_i)} +\right)\right|_{Ξ΅=0}\\ +&= \left( + \frac{\left(2 e_i^H A (x + Ξ΅e_i) \right) \left( (x + Ξ΅e_i)^H (x + Ξ΅e_i) \right) } + { \left( (x + Ξ΅e_i)^H (x + Ξ΅e_i) \right)^2 } \right. \\ + &\quad\left.\left. -\frac{ \left( 2 e_i^H (x + Ξ΅e_i) \right) \left( (x + Ξ΅e_i)^H A (x + Ξ΅e_i) \right)} + { \left( (x + Ξ΅e_i)^H (x + Ξ΅e_i) \right)^2 } +\right) \right|_{Ξ΅=0}\\ +&= 2\frac{e_i^H A x}{x^Hx} - 2 \frac{R_A(x)}{x^H x} e_i^H x\\ +&= \frac{2}{x^H x} \big( (A x)_i - R_A(x) \, x_i \big) +\end{aligned} +``` +such that +```math +\nabla R_A(x) = 2 \frac{Ax - R_A(x) x}{\| x \|^2}. +``` +We note that the residual $r = Ax - R_A(x) x$ points along the direction of the gradient of $R_A(x)$. + +A classic algorithm to solve such minimisation problems in the objective $R_A(x)$ is **preconditioned gradient descent**, in which the iterates are related as +```math +x^{(i+1)} \leftarrow x^{(i)} - \tilde{Ξ±} P^{-1} \nabla R_A(x^{(i)}) +``` +i.e. in which simply a scaled fraction of the gradient is removed from the current iterate. In our case this becomes +```math +x^{(i+1)} \leftarrow x^{(i)} - Ξ± P^{-1} r^{(i)} = x^{(i)} - Ξ± P^{-1} \left( A x^{(i)} - R_A(x^{(i)}) \, x^{(i)} \right), +``` +where $r^{(i)} = Ax^{(i)} - R_A(x^{(i)}) x^{(i)}$. + +In practice the step-size $Ξ±$ and the preconditioner matrix $P^{-1}$ need to be chosen cleverly to obtain fast convergence. Algorithmically we can write: +""" + +# ╔═║ 1d099205-9d84-4ecc-a2af-81910326a4ba +function preconditioned_gradient_descent(A; x=randn(eltype(A), size(A, 2)), + Ξ±=1.0, Pinv=I, tol=1e-6, maxiter=100, + verbose=true) + T = real(eltype(A)) + + eigenvalues = T[] + residual_norms = T[] + Ξ» = NaN + for i in 1:maxiter + normalize!(x) + Ax = A * x + Ξ» = x'Ax # Rayleigh quotient + r = Ax - Ξ» * x # Residual + norm_r = norm(r) + + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ» norm_r + push!(eigenvalues, Ξ») + push!(residual_norms, norm_r) + norm_r < tol && break + + # Precondition the residual and move along the direction: + x = x - Ξ±Β * Pinv * r + end + converged = residual_norms[end] < tol + !converged && verbose && @warn "PGD not converged." + + (; Ξ», x, converged, eigenvalues, residual_norms) +end + +# ╔═║ f17500cc-6e5f-442d-9b22-2dc406367ad6 +Apgd = Diagonal(abs.(randn(100))) + +# ╔═║ 320a4138-1d2d-4039-adaa-1aeb7f1929cf +let + Pinv = Diagonal(1 ./ diag(Apgd)) # Diagonal preconditioner for Apgd + (; residual_norms) = preconditioned_gradient_descent(Apgd; Pinv) + plot(residual_norms; yaxis=:log, label="prec. grad. desc. residual λ₁", lw=2, mark=:x) +end + +# ╔═║ db913617-2b81-4ea7-a907-033dd31bda5a +md"Convergence is thus great if a good preconditioner is chosen. However, for no preconditioner we obtain:" + +# ╔═║ 68fff309-a3ec-4ebc-bcfe-bafeac6ef3f5 +preconditioned_gradient_descent(Apgd) + +# ╔═║ f4bb1d69-1f06-4769-ac6e-081ddaa437d7 +md""" +**Preconditioned gradient descent** is **closely related to the inverse power method**. Namely, if we chose $P^{-1} = A^{-1}$ and $Ξ± = 1$, then we obtain +```math +x^{(i+1)} \leftarrow x^{(i)} - A^{-1} r^{(i)} = x^{(i)} - A^{-1} \left( A x^{(i)} - R_A(x^{(i)}) \, x^{(i)} \right) += R_A(x^{(i)}) A^{-1} x^{(i)}, +``` +which is a scaled version of the inverse power method. +If one considered preconditioned gradient ascent one could +similarly draw a connection to the power method. +""" + +# ╔═║ 8481465b-171b-4374-8ec9-d7c19bd23d81 +md""" +### Choice of preconditioner + +What preconditioner should be chosen is often highly problem dependent. However, there are a number of standard choices, which can often be employed blindly to speed up the calculations. Popular are in particular incomplete factorisations (e.g. Cholesky) or lower-precision version of the original problem. A detailed discussion is out of the scope of these notes. Details can be found, for example, in book of Saad. + +""" + +# ╔═║ 4b0cd6fc-50ec-4e12-82e0-30fe2ceb98ce +md""" +### Step sizes and LOBPCG + +The final aspect we want to briefly address is **how to choose the step size** $Ξ±$. The best step size can be found by a **line search**, +i.e. by solving the one-dimensional optimisation problem +```math +Ξ±_{i,\text{ls}} = \text{argmin}_{Ξ±\in \mathbb{R}} R_A(x^{(i)} + Ξ± \tilde{r}^{(i)}) +``` +where we denoted $\tilde{r}^{(i)} = P^{-1} r^{(i)}$. +The next iterate is then formed +as $x^{(i+1)} = x^{(i)} + Ξ±_{i,\text{ls}} \tilde{r}^{(i)}$. +Since the Rayleigh quotient contains normalisation in the denominator +we note that +```math +R_A(Ξ΄ x^{(i)} + Ξ΅ \tilde{r}^{(i)} ) = R_A( x^{(i)} + \frac{Ξ΅}{Ξ΄} \tilde{r}^{(i)} ). +``` +Therefore, our full procedure could equivalently be written as +```math +x^{(i+1)} = \text{argmin}_{y \in \text{span} \left\{ x^{(i)}, \tilde{r}^{(i)} \right\}} R_A(y), +``` +i.e. as a **minimisation in the subspace spanned** by $x^{(i)}$ and $\tilde{r}^{(i)}$. +""" + +# ╔═║ 9fdc92f6-158f-4e62-82e0-b6c5ce96d9a8 +md""" +A **slight generalisation** of this idea is to make the minimisation **subspace even larger** by also including the *previous* iterate to the search space, i.e. to employ +```math +x^{(i+1)} = \text{argmin}_{y \in \text{span} \left\{ x^{(i)}, x^{(i-1)}, \tilde{r}^{(i)} \right\}} R_A(y). \qquad \text{$(\ast)$} +``` +This turns out to improve the overall observed convergence rates considerably. This modification has been suggested following a more detailed investigation of minimisation approaches for eigenvalue problems, in particular a comparison with more sophisticated minimisation algorithms such as the conjugate gradient (CG) approach, which is out of the scope of this lecture. + +The improved convergence rates of $(\ast)$ unfortunately comes with a downside: As the iterations converge $x^{(i)}$ and $x^{(i-1)}$ *both* converge to the exact eigenvector, such that $x^{(i)}$ and $x^{(i-1)}$ become more and more **linearly dependent**, which makes the optimisation unstable. This is in practice circumvented by two further modifications: +- Instead of using the basis vectors $\{x^{(i)}, x^{(i-1)}, \tilde{r}^{(i)}\}$ to build the subspace, we compute $p^{(i)} = x^{(i)} - x^{(i-1)}$ + and instead employ $\{ x^{(i)}, p^{(i)}, \tilde{r}^{(i)} \}$, + which span the same space. +- Instead of first performing the line search to find the optimal $x^{(i+1)}$ and then computing the Rayleigh coefficient $R_A(x^{(i+1)})$, we perform a **Rayleigh-Ritz procedure projecting onto the subspace** $\mathcal{S} = \text{span} \{ x^{(i)}, p^{(i)}, \tilde{r}^{(i)} \}$. Due to our results on subspace methods (in particular Proposition 5.5.) we know that a Rayleigh-Ritz in the subspace $\mathcal{S}$ is equivalent to finding the minimiser of $R_A(x)$ where $x \in \mathcal{S}$. + +These two modifications with our previous discussion lead to the **LOPCG (locally optimal preconditioned conjugate gradient)** algorithm: +""" + +# ╔═║ 94e6808b-c89c-4431-90d8-5e252e38d834 +function lopcg(A; x=randn(eltype(A), size(A, 2)), ortho=ortho_qr, + Pinv=I, tol=1e-6, maxiter=100, verbose=true) + T = real(eltype(A)) + + eigenvalues = T[] + residual_norms = T[] + Ξ» = NaN + p = nothing + r = nothing + + for i in 1:maxiter + if i > 1 + Z = hcat(x, p, r) + else + Z = hcat(x) + end + Z = ortho(Z) + + # Rayleigh-Ritz step to get smallest eigenvalue + AZ = A * Z + Ξ», Y = eigen(Hermitian(Z' * AZ)) + Ξ» = Ξ»[1] # Keep only smallest eigenvalue + y = Y[:, 1] # Keep only smallest eigenvector + new_x = Z * y # x^{(i)} is the eigenvector of the Rayleigh-Ritz, + # since this gives the optimal linear combination between + # p^{(i-1)}, x^{(i-1)} and Μƒr^{(i-1)} + + # Store results and residual + push!(eigenvalues, Ξ») + r = AZ * y - new_x * Ξ» + norm_r = norm(r) + push!(residual_norms, norm_r) + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ» norm_r + if norm_r < tol + x = new_x + break + end + + # Precondition residual, update x and p + r = Pinv * r + p = x - new_x + x = new_x + end + converged = residual_norms[end] < tol + !converged && verbose && @warn "LOPCG not converged." + + (; Ξ», x, converged, eigenvalues, residual_norms) +end + +# ╔═║ f1398a23-90d9-40ea-8ef2-ebcd0ab67b14 +Alopcg = Diagonal(abs.(randn(1000)).^0.1); + +# ╔═║ bba31f5b-7eca-412b-99c7-65adcfa2be1b +let + Pinv = Diagonal(1 ./ diag(Alopcg)) # Diagonal preconditioner for Apgd + (; residual_norms) = lopcg(Alopcg; Pinv) + + plot(residual_norms; yaxis=:log, lw=2, label="LOPCG residual norms", mark=:x) +end + +# ╔═║ 9b7bdce7-38cb-4c7c-b1b4-d1ec3c9fa168 +md""" +The generalisation to an algorithm capable of computing multiple eigenpairs is straightforward, leading to **LOBPCG (locally optimal *block* preconditioned conjugate gradient)** is only bookkeeping: +""" + +# ╔═║ 42777c48-0f3a-4c43-974b-7320ad16efb9 +function lobpcg(A; X=randn(eltype(A), size(A, 2), 2), ortho=ortho_qr, + Pinv=I, tol=1e-6, maxiter=100, verbose=true) + T = real(eltype(A)) + m = size(X, 2) # block size + + eigenvalues = Vector{T}[] + residual_norms = Vector{T}[] + Ξ» = NaN + P = nothing + R = nothing + + for i in 1:maxiter + if i > 1 + Z = hcat(X, P, R) + else + Z = X + end + Z = ortho(Z) + + # Rayleigh-Ritz step to get smallest eigenvalues + AZ = A * Z + Ξ», Y = eigen(Hermitian(Z' * AZ)) + Ξ» = Ξ»[1:m] + Y = Y[:, 1:m] + new_X = Z * Y + + # Store results and residual + push!(eigenvalues, Ξ») + R = AZ * Y - new_X * Diagonal(Ξ») + norm_r = norm.(eachcol(R)) + push!(residual_norms, norm_r) + verbose && @printf "%3i %8.4g %8.4g\n" i Ξ»[end] norm_r[end] + if maximum(norm_r) < tol + X = new_X + break + end + + # Precondition residual, update X and P + R = Pinv * R + P = X - new_X + X = new_X + end + + (; Ξ», X, eigenvalues, residual_norms) +end + +# ╔═║ 05175611-34de-44cf-a385-c493a6f3cf19 +let + Pinv = Diagonal(1 ./ diag(Alopcg)) # Diagonal preconditioner for Apgd + (; residual_norms) = lobpcg(Alopcg; Pinv) + + norms_first = [n[1] for n in residual_norms] + norms_second = [n[2] for n in residual_norms] + + p = plot(; yaxis=:log) + plot!(p, norms_first, label="residual first eigenpair", c=1, lw=2, mark=:x) + plot!(p, norms_second, label="residual second eigenpair", c=2, lw=2, mark=:x) +end + +# ╔═║ 15220d0d-2c9f-4daf-a290-672ca439faec +md""" +Note, that this simple implementation of LOBPCG is **still not numerically stable** and **considerable issues remain**. In practice one should thus always employ one of the well-tested implementation in standard numerical libraries. + +For example the LOBPCG of the DFTK package: +""" + +# ╔═║ 23990899-05d2-4b40-9d52-23caf218c7cf +let + # Note that DFTK wants the operator P and P⁻¹ + prec = Diagonal(diag(Alopcg)) + + X = randn(size(Alopcg, 2), 2) + (; residual_history) = DFTK.lobpcg_hyper(Alopcg, X; + prec, tol=1e-6, display_progress=true) + + residual_history .+= eps(Float64) # to ensure we can use logarithmic axes + + p = plot(; yaxis=:log, ylims=(1e-7, Inf)) + plot!(p, residual_history[1, :], label="residual first eigenpair", + c=1, lw=2, mark=:x) + plot!(p, residual_history[2, :], label="residual second eigenpair", + c=2, lw=2, mark=:x) +end + +# ╔═║ 461162d6-b0e6-41c4-8e60-6293d2fc25ee +md""" +## How do the methods compare ? + +To summarise our survey, we want to briefly compare the methods on an example. +We set up a test problem, where we can tune both the size of the matrix and +the gaps between the first eigenvalues. We then try multiple methods to obtain the **smallest** one or two eigenpairs: +""" + +# ╔═║ 92a6feb4-88ad-4681-b6dd-2df2a790360c +md""" +### Single-vector approaches +""" + +# ╔═║ aa9f8c4b-b733-48c6-93f0-4ab8d52b2a3e +md""" +- `logN_s`: Logarithm of the **problem size** $(@bind logN_s PlutoUI.Slider(1.2:0.1:4.0, default=2.0)) +- `log1_s`: Log of gap between **1st and 2nd** eigenvalue: $(@bind log1_s PlutoUI.Slider(-6:0.1:3.0; show_value=true, default=1.0)) +- `logPrec_s`: **Preconditioner noise** level $(@bind logPrec_s PlutoUI.Slider(-3:0.1:-1.5, default=-2.5, show_value=true)) + +Plotting options: +- `maxiter_s`: Plotting range is from `0` to `maxiter`: $(@bind maxiter_s PlutoUI.Slider(10:1:100; show_value=true, default=20)) +""" + +# ╔═║ 99a01adc-34e2-4284-b918-4bfc020b47d6 +begin + N_s = ceil(Int, 10^logN_s) + A_s = Diagonal([1.0 + 1.0 + 10^log1_s]) + B_s = Diagonal(A_s[2, 2] .+ 10 .+ randn(N_s-2)) + O_s = zeros(2, N_s-2) + + M_s = [A_s O_s; + O_s' B_s] +end + +# ╔═║ 1cf09a9d-2e2b-4a0c-848c-56d3bd1e7f87 +x = randn(size(M_s, 2)); # Consistent starting vector for all methods + +# ╔═║ 39215879-264c-4c2a-bafc-a792abf70017 +Pnoise_s = 10^logPrec_s * randn(size(M_s, 1)); + +# ╔═║ ff50c8b6-ce84-4d74-b5ba-f5abecdd7fb2 +let + function inverse_power_method(A; kwargs...) + A⁻¹ = InverseMap(factorize(A)) + power_method(A⁻¹; kwargs...) + end + + p = plot(yaxis=:log, ylims=(1e-7, 10)) + + non_preconditioned_methods = ( + inverse_power_method, rayleigh_quotient_iteration, + ) + for method in non_preconditioned_methods + (; residual_norms) = method(M_s; x, verbose=false, tol=1e-6) + plot!(p, residual_norms, label=string(method), lw=2, mark=:x) + end + + preconditioned_method = ( + preconditioned_gradient_descent, lopcg + ) + for (i, method) in enumerate(preconditioned_method) + c = i + length(non_preconditioned_methods) + + # Perfect preconditioner: The inverse diagonal + Pinv = Diagonal(1 ./ diag(M_s)) + (; residual_norms) = method(M_s; x, verbose=false, tol=1e-6, Pinv) + plot!(p, residual_norms; label=string(method) * " (perfect precon)", + lw=2, c, mark=:x) + + # Preconditioner plus noise + Pinv = Diagonal(1 ./ diag(M_s) .+ Pnoise_s) + (; residual_norms) = method(M_s; x, verbose=false, tol=1e-6, Pinv) + plot!(p, residual_norms; label=string(method) * " (noisy precon)", + lw=2, c, ls=:dash, mark=:x) + end + + default_lim = xlims(p) + xlims!(p, 0, min(default_lim[2], maxiter_s)) +end + +# ╔═║ f9e4e29c-247d-4896-90d0-9c5f716c203c +md"""By playing with the sliders you should notice: + +- **Rayleigh quotient iteration** generally performs **best**, but is also the most expensive (Full factorisation required at *every iteration*) +- **Inverse power method** and **preconditioned gradient descent fail** when the gap becomes **too small**. +- Noise in the preconditioner leads to worse convergence in LOPCG and preconditioned gradient descent, but LOPCG is more resistant +- Overall LOPCG offers a pretty good compromise. +""" + +# ╔═║ 2f4000e9-eaca-446d-af43-3b8ea06eb2b1 +md""" +### Block and subspace approaches +""" + +# ╔═║ fb887ea1-b4bb-49d5-af84-9a1278791d67 +md""" +- `logN_b`: Logarithm of the **problem size** $(@bind logN_b PlutoUI.Slider(1.2:0.1:4.0, default=2.0)) +- `log1_b`: Log of gap between **1st and 2nd** eigenvalue: $(@bind log1_b PlutoUI.Slider(-6:0.1:3.0; show_value=true, default=1.0)) +- `log2_b`: Log of gap between **2nd and 3rd** eigenvalue: $(@bind log2_b PlutoUI.Slider(-6:0.1:3.0; show_value=true, default=1.0)) +- `logPrec_b`: **Preconditioner noise** level $(@bind logPrec_b PlutoUI.Slider(-3:0.1:-1.5, default=-2.5, show_value=true)) + +Plotting options: +- `maxiter_b`: Plotting range is from `0` to `maxiter`: $(@bind maxiter_b PlutoUI.Slider(10:1:100; show_value=true, default=20)) +""" + +# ╔═║ 9d86090f-9680-43fa-b86b-2fc664209faf +begin + N_b = ceil(Int, 10^logN_b) + A_b = Diagonal([1.0 + 1.0 + 10^log1_b + 1.0 + 10^log1_b + 10^log2_b]) + B_b = Diagonal(A_b[3, 3] .+ 10 .+ randn(N_b-3)) + O_b = zeros(3, N_b-3) + + M_b = [A_b O_b; + O_b' B_b] +end + +# ╔═║ eb909306-b913-464b-81d6-becc9946e7c1 +X = randn(size(M_b, 2), 2); # Consistent starting vectors for all methods + +# ╔═║ aa4f358a-e8fa-4258-bcb6-b6707ef62b0a +Pnoise_b = 10^logPrec_b * randn(size(M_b, 1)); + +# ╔═║ bdcc33c7-eda7-48fe-a73c-84c2e7c64dab +let + function lobpcg_dftk(A; X, Pinv::Diagonal, tol, verbose) + prec = Diagonal(1 ./ diag(Pinv)) + (; Ξ», residual_history) = DFTK.lobpcg_hyper(A, X; prec, tol, + display_progress=verbose) + residual_norms = [Array(c) for c in eachcol(residual_history)] + (; Ξ», residual_norms) + end + + function plot_both!(p, Ξ», residual_norms; label, kwargs...) + # Extract residual, but make sure that eigenpairs are sorted ascendingly + sorting = sortperm(Ξ») + resid_1st = [resid[sorting[1]] for resid in residual_norms] + resid_2nd = [resid[sorting[2]] for resid in residual_norms] + + plot!(p, resid_1st; label=label * " λ₁", m=:x, lw=2, markersize=4, kwargs...) + plot!(p, resid_2nd; label=label * " Ξ»β‚‚", m=:+, lw=2, markersize=4, kwargs...) + end + + p = plot(yaxis=:log, ylims=(1e-7, 10)) + c = 1 + + for method in (subspace_iteration, projected_subspace_iteration) + M⁻¹ = Diagonal(1 ./ diag(M_b)) + (; Ξ», residual_norms) = method(M⁻¹; X, verbose=false, tol=1e-6) + plot_both!(p, 1 ./ Ξ», residual_norms; label=string(method), c) + c += 1 + end + + for method in (lobpcg, lobpcg_dftk) + # Perfect preconditioner: The inverse diagonal + Pinv = Diagonal(1 ./ diag(M_b)) + (; Ξ», residual_norms) = method(M_b; X, verbose=false, tol=1e-6, Pinv) + plot_both!(p, Ξ», residual_norms; label=string(method) * "(perfect precon)", c) + + # Preconditioner plus noise + Pinv = Diagonal(1 ./ diag(M_b) .+ Pnoise_b) + (; Ξ», residual_norms) = method(M_b; X, verbose=false, tol=1e-6, Pinv) + plot_both!(p, Ξ», residual_norms; label=string(method) * "(noisy precon)", + c, ls=:dash) + + c += 1 + end + + default_lim = xlims(p) + xlims!(p, 0, min(default_lim[2], maxiter_b)) +end + +# ╔═║ 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\ No newline at end of file diff --git a/07_Perturbation_theory.jl b/07_Perturbation_theory.jl new file mode 100644 index 0000000..b0d95c8 --- /dev/null +++ b/07_Perturbation_theory.jl @@ -0,0 +1,1369 @@ +### A Pluto.jl notebook ### +# v0.20.3 + +using Markdown +using InteractiveUtils + +# ╔═║ 32b2d18c-edc0-11ee-13d9-83a8a646b6bd +begin + import TikzPictures.TikzPicture + using LaTeXStrings + using PlutoUI + using HypertextLiteral + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ 57032426-2ff0-442d-aa5b-37a7c621d83f +md""" +# Matrix Perturbation Theory +""" + +# ╔═║ 25758608-0d58-474e-b6b1-d3463938d1dc +md""" +We saw in the previous lectures how to find the eigenpairs $(\tilde{\lambda}_{i}, \tilde{x}_{i} )$ of a Hermitian matrix $\tilde{A} \in \mathbb{C}^{n \times n}$. +However, in our discussion about floating point numbers we already noted that sometimes $A$ itself cannot be represented exactly, such that effectively we find the eigenpairs of $\tilde{A}+\Delta A$, where $\Delta A \in \mathbb{C}^{n \times n}$ is a small error. +The same happens if we deliberately employ reduced precision or approximate the computation of $\tilde{A}$ itself. +A natural question is thus : +- How do the eigenpairs change if we go from $A$ to $\tilde{A}+\Delta A$ ? +One way to track this change in $\tilde{\lambda}_{i}$ and $\tilde{x}_{i}$ approximately without recomputing the eigenpairs of $\tilde{A}+\Delta A$ from scratch is **perturbation theory**. + +Our setup is as follows: +- We have available, for the Hermitian matrix $\tilde{A} \in \mathbb{C}^{n \times n}$, an approximate set of eigenpairs $(\tilde{\lambda}_{i}, \tilde{x}_{i})$, and we want to find a correction to the lowest few eigenpairs, such that they are approximately eigenpairs of $A(t)=\tilde{A}+t \Delta A$ where $t \in \mathbb{C}$ and $\Delta A \in \mathbb{C}^{n \times n}$ Hermitian. +- We care in particular about the regime $t \simeq 0$. +- Clearly the eigenpairs of $A(t)$ are also $t$-dependant with $\lambda_{i}(t=0)=\tilde \lambda_{i}, x_{i}(t=0)=\tilde x_{i}$. +- It would be natural to just expand $\lambda_{i}$ and $x_{i}$ in a Maclaurin series of $t$. + However, it is not clear *a priori* whether this series converges, what its convergence radius is and thus if this would even make sense. +- We will thus first study linear perturbations $\tilde A + t\Delta A$, for which we first need the notion of a resolvent. + +""" + +# ╔═║ 490b2be8-78e5-46a3-b313-413f90f8265d +md""" +## Resolvent + +!!! note "Definition (Resolvent set and resolvent)" + Let $A \in \mathbb{C}^{n \times n}$ and $z \in \mathbb{C}$. The set + ```math + \resolvent(A)=\{z \in \mathbb{C} \mid A-zI \text{ is invertible} \} + ``` + is called the **resolvent set** of $A$. + For $z \in \rho(A)$ the transformation + ```math + R_{z}(A)=(A-z I)^{-1} + ``` + is called the **resolvent** of $A$. + +!!! tip "Remark" + In order to distinguish the resolvent set from the spectral radius, both used in this chapter and denoted by "rho", we will use $\resolvent$ (`\rho`) for the resolvent set and $\spectralradius$ (`\varrho`) for the spectral radius. +""" + +# ╔═║ ad786ef1-0b62-4a9a-84ca-c776d95e7cbc +md""" +!!! tip "Remark" + - The mapping $z \mapsto R_{z}(A)$ has singularities exactly at the eigenvalues of $A$. In fact + ```math + \sigma(A)=\mathbb{C} \setminus \resolvent(A) + ``` + is one way to define the **spectrum** of $A$. + + - Anywhere else $z \mapsto R_{z}(t)$ is analytic. + Indeed, we have for any $\varepsilon>0$ and any $z$ from an $\varepsilon$-ball around $z_0$ away from eigenvalues, i.e. $z \in B_{\varepsilon} (z_{0}) \subset \resolvent(A)$, that + ```math + \begin{aligned} + R_{z}(A)=(A-z)^{-1} & =\left[\left(A-z_{0} I\right)-\left(z-z_{0}\right) I\right]^{-1} \\ + & =R_{z_{0}}(A)\left[I-\left(z-z_{0}\right) R_{z_{0}}(A)\right]^{-1} + \end{aligned} + ``` + Now $\left[I-\left(z-z_{0}\right) R_{z_{0}}(A)\right]^{-1}$ can be expanded into a Neumann series + ```math + \left[I-\left(z-z_{0}\right) R_{z_{0}}(A)\right]^{-1}=\sum_{k=0}^{\infty}\left[\left(z-z_{0}\right) R_{z_{0}}(A)\right]^{k} + ``` + whenever the spectral radius of $\left(z-z_{0}\right) R_{z_{0}}(A)$ is less the unity. + The Taylor expansion of $R_{z}(A)$ thus exists as long as $z \in B_{\varepsilon} (z_{0} )$ with $\varepsilon<1 / \spectralradius \left(R_{z_{0}}(A)\right)$ and has the form + + ```math + R_{z}(A)=\sum_{k=0}^{\infty}\left(z-z_{0}\right)^{k} R_{z_{0}}(A)^{k+1} + ``` + - A consequence (use Cramer's rule) is that the singularities at $z=\lambda_{1}, \dots, \lambda_{n}$ are not essential. + Thus, the resolvent is **meromorphic** on $z$ (only has a countable number of singularities, which are removable). + +""" + +# ╔═║ d5a28a85-68f8-45de-912b-6944edbf1de7 +md""" + +We also mention the resolvent identities, which will prove useful later : +```math +\begin{align} + R_{z_{1}}(A)-R_{z_{2}}(A) &= \left(z_{1}-z_{2}\right) R_{z_{1}}(A) R_{z_{2}}(A) + \tag{1st identity} + \\ + R_{z}(A)-R_{z}(B)&=R_{z}(A)(B-A) R_{z}(B) + \tag{2nd identity} +\end{align} +``` +The first identity is also known as the Hilbert identity. + +In what follows we emphasize the use of the resolvent to obtain spectral information + + +""" + +# ╔═║ 9845a2db-1878-435d-bb4f-e8e77cbde85d +md""" +## Spectral projectors + +Let $\lambda_{i}$ be $a_{n}$ eigenvalue of $A$ +and let $\contour_{\lambda_{i}}$ be a counter-clockwise contour enclosing no other eigenvalues of $A$, as illustrated below. +""" + +# ╔═║ c3fd5d2f-9907-4500-8f40-511752fc6238 +TikzPicture(L""" +%Shape: Polygon Curved [id:ds06447437278331569] +\draw[blue] (119,73.37) .. controls (143,57.37) and (154.03,80.73) .. (168.5,80.87) node{>} .. controls (182.97,81.01) and (196.5,58.87) .. (217.5,58.37) node{>} .. controls (238.5,57.87) and (242.5,70.37) .. (242,93.37) node{$ +\wedge$} .. controls (241.5,116.37) and (216,128.87) .. (197,129.37) node{<} node[above]{$C_{\lambda_i}$} .. controls (178,129.87) and (159,125.37) .. (140,133.87) .. controls (121,142.37) and (91.83,115) .. (123,110.87) .. controls (154.17,106.73) and (152.62,102.92) .. (150.5,99.87) .. controls (148.38,96.81) and (95,89.37) .. (119,73.37) -- cycle ; + +\draw[->] (50,100) -- node{$\times$} node[above]{$\lambda_{i-1}$} (200,100) node{$\times$} node[above]{$\lambda_{i}$} -- node{$\times$} node[above]{$\lambda_{i+1}$} (300,100) ; +""", width="23cm", options="scale=0.05", preamble=raw"\usepackage{amsfonts}") + +# ╔═║ 7f02b176-3057-4ed6-8ebe-31d645f16e48 +md""" +We consider the integral +```math +P_{i}=-\frac{1}{2 \pi i} \oint_{\contour_{\lambda_{i}}} R_{z}(A) d z +``` +and establish these are orthogonal projectors for $i \neq j$ : +""" + +# ╔═║ e2f62f58-6614-4b80-b225-9e3136027f8d +md""" +!!! note "Theorem 1" + The linear transformations $P_{i}, i=1, \dots, n$ defined above associated with distinct eigenvalues $i=1, \dots, n$ are such that + + - (a) $P_{i}^{2}=P_{i}$ + + - (b) $P_{i} P_{j}=P_{j} P_{i}=0$ if $i \neq j$ + + - (c) $\sum_{i=1}^{n} P_{i}=I$ +""" + +# ╔═║ 2210e9c0-53e3-472e-80e7-4151d14eec9d +md""" +> *Proof* of (a). +> Let $\contour_{\lambda_{i}}$ and $\contour_{\lambda_{i}}^{\prime}$ be two curves enclosing $\lambda_{i}$ with $\contour_{\lambda_{i}}^{\prime}$ enclosing $\contour_{\lambda_{i}}$ (illustrated below). Then +>```math +> \begin{aligned} +> (2 i \pi)^{2} P_{i}^{2} & =\oint_{\contour_{\lambda_{i}}} \oint_{\contour_{\lambda_{i}}^{\prime}} R_{z}(A) R_{z^{\prime}}(A) d z d z^{\prime} \\ +> & \hspace{-1.3em} \stackrel{\text{1st identity}}{=} \oint_{\contour_{\lambda_{i}}} \oint_{\contour_{\lambda_{i}^{\prime}}} \frac{1}{z^{\prime}-z}\left(R_{z^{\prime}}(A)-R_{z}(A)\right) d z dz' +> \end{aligned} +>``` +>Now we use Cauchy's integral formula to obtain +>```math +> \oint_{\contour_{\lambda_{i}}} \frac{d z}{z^{\prime}-z}=0 +>``` +>since $z'$ is outside of $\contour_{\lambda_i}$ and thus there are no singularities inside $\contour_{\lambda_i}$, and +>```math +> \oint_{\contour_{Ξ»_i}^{\prime}} \frac{d z^{\prime}}{z^{\prime}-z}=2 \pi i +>``` +>since $z$ is inside $\contour_{\lambda_i}'$ +>Therefore, +> ```math +> \begin{align} +> & \oint_{\contour_{\lambda_{i}}} \oint_{\contour_{\lambda_{i}}^{\prime}} \frac{R_{z^\prime}\left(A\right)}{z^{\prime}-z} d z^{\prime} d z=\oint_{\contour_{\lambda_{i}}^{\prime}} R_{z^\prime}\left(A\right)\left(\oint_{\contour_{\lambda_{i}}} \frac{d z}{z^{\prime}-z}\right) d z^{\prime}=0 +> \\ +> & \oint_{\contour_{\lambda_{i}}} \oint_{\contour_{\lambda_i}^{\prime}} \frac{R_z(A)}{z^{\prime}-z} d z^{\prime} d z=\oint_{\contour_{\lambda_i}} R_z(A)\left(\oint_{\contour_{\lambda_{i}}^{\prime}} \frac{d z^{\prime}}{z^{\prime}-z}\right) d z = 2 \pi i \oint_{\contour_{\lambda_i}} R_z(A) dz +> \end{align} +> ``` +> Such that $P_{i}^{2}=P_{i}$. $\hspace{12cm} \square$ +""" + +# ╔═║ 38b1496f-cc7f-47c7-b40f-119d2dd1e663 +TikzPicture(L""" +%Rounded Rect [id:dp03377449697776913] +\draw [draw opacity=0][fill={rgb, 255:red, 242; green, 242; blue, 242 } ,fill opacity=1 ] (20,45.5) .. controls (20,41.36) and (23.36,38) .. (27.5,38) -- (232.5,38) .. controls (236.64,38) and (240,41.36) .. (240,45.5) -- (240,147.37) .. controls (240,151.51) and (236.64,154.87) .. (232.5,154.87) -- (27.5,154.87) .. controls (23.36,154.87) and (20,151.51) .. (20,147.37) -- cycle ; + +%Shape: Rectangle [id:dp6341955401888753] (buffer) +\draw [draw opacity=0] (9.5,45.87) -- (20,45.87) -- (20,147.37) -- (9.5,147.37) -- cycle ; + + +%Shape: Polygon Curved [id:ds8789544884464016] +\draw[blue] (124,50.37) .. controls (158.5,50.37) and (187.5,70.87) .. (187,99.87) node{$\wedge$} .. controls (186.5,128.87) and (166.5,137.37) .. (148,139.87) node{$\times$} node[above right]{$z'$} .. controls (129.5,142.37) and (83,145.5) .. (64.5,109.87) node[left]{$C_{\lambda_i}'$} .. controls (46,74.23) and (89.5,50.37) .. (124,50.37) -- cycle ; + +%Shape: Polygon Curved [id:ds35740336827854424] +\draw[red] (126.5,71.37) .. controls (149,71.87) and (168.5,83.37) .. (168,100.87) node{$\wedge$} .. controls (167.5,118.37) and (154,120.37) .. (142.5,123.37) node{$\times$} node[above right]{$z$} .. controls (131,126.37) and (102.5,129.37) .. (90.5,108.37) node[left]{$C_{\lambda_i}$} .. controls (78.5,87.37) and (104,70.87) .. (126.5,71.37) -- cycle ; + +\draw[->] (30,90) -- node{$\times$} node[above]{$\lambda_i$} (230,90); +""",width="20cm",options="scale=0.04",preamble=raw"\usepackage{amsfonts}") + +# ╔═║ aefedae9-b792-41db-bb52-c8540619bcf1 +md""" +> *Proof* of (b). +> The proof follows a very similar argument since $\contour_{\lambda_{i}}$ and $\contour_{\lambda_{j}}$ can be taken to be non-intersecting. +""" + +# ╔═║ 36f4bb7f-e1d5-4587-b96a-574820601963 +md""" +> *Proof* of (c). +>We consider +>```math +> P=-\frac{1}{2 \pi i} \sum_{i=1}^{n} \oint_{\contour_{\lambda_{i}}} R_z(A) d z . +>``` +> $R_z(A)$ has no poles apart from the eigenvalues $\{\lambda_{i}\}_{i=1}^n$, so we can replace the sum over integrals enclosing single eigenvalues by a single integral enclosing all eigenvalues. +>We choose this curve to be a circle $C$ with radius $r$ about the origin. +>Thus +>```math +> P=-\frac{1}{2 \pi i} \oint_{C} R_z(A) d z +>``` +>By a change of variables $t=1 / z$ : +>```math +> P=-\frac{1}{2 \pi i} \oint_{\contour_{-}^{\prime}}\left(A- \frac1{t} I\right)^{-1}\left(-\frac{d t}{t^{2}}\right)=-\frac{1}{2 \pi i} \oint_{\contour_{+}^{\prime}}(t A-I)^{-1} \frac{d t}{t} +>``` +>where $\contour_{+/-}^{\prime}$ are the circles about the origin with radius $1/r$ that run counter-clockwise/clockwise. In the second step we have absorbed a negative sign into changing the direction of the contour from counter-clockwise to clockwise. +>Note that $r$ must be larger than the spectral radius of $A$, $\spectralradius(A)$, thus $\spectralradius(t A)<1$. +>Therefore, $(I-t A)^{-1}$ can be expanded into a Neumann series +>```math +> (I-t A)^{-1}=\sum_{k=0}^{\infty}(t A)^{k} +>``` +>which converges, from which +>```math +> \begin{align} +> P & =\frac{1}{2 \pi i} \oint_{\contour_{+}^{\prime}}\left[\sum_{k=0}^{\infty} t^{k-1} A^{k}\right] d t && \\ +> & \hspace{-1.5em} \stackrel{\text{Residue Thm}}{=} \frac{1}{2 \pi i} \oint_{\contour_{+}^{\prime}} \frac{I}{t} d t +> \\ +> & \hspace{-1.5em} \stackrel{\text{Residue Thm}}{=\def\laplacian{{\Delta}}} \frac{2 \pi i}{2 \pi i} I=I +> && \square +> \end{align} +>``` +""" + +# ╔═║ 129c7fb1-479e-4e82-a17c-0b153d2d890d +md""" +!!! note "Theorem 2" + Let $A \in \mathbb{C}^{n \times n}$ Hermitian. + We have + ```math + \im \left(P_{i}\right)=\ker\left(A-\lambda_{i} I\right)=\eigenspace_A\left(\lambda_{i}\right) + ``` + i.e. the image of the projector $P_{i}$ associated to $\lambda_{i}$ is an invariant subspace of $A$, which is equal to the eigenspace associated to $\lambda_{i}$. +""" + +# ╔═║ fec3438a-8da1-4d10-9b71-7c7b87d08fdf +md""" +> *Proof.* +> First note that $R_{z}(A)$ and $A$ commute since +> ```math +> \begin{aligned} +> R_z(A)\, A &= (A - z I)^{-1} (A^{-1})^{-1} = \left( A^{-1} A - A^{-1}z \right)^{-1} \\ &= \left( A A^{-1} - z A^{-1} \right)^{-1} = A (A - zI)^{-1} = A \, R_z(A) +> \end{aligned} +> ``` +> we get by integration $P_{i} A=A P_{i}$, such that $\im \left(P_{i}\right)$ is invariant under $A$. +> +>Next we show $\im \left(P_{i}\right) \supseteq \ker\left(A-\lambda_{i} I\right)$. +>Consider an $x \in \ker\left(A-\lambda_{i} I\right)$ and expand +>```math +> \begin{aligned} +> R_{z}(A) x & =(A-z I)^{-1} x \\ +> & =\left[\left(A-\lambda_{i} I\right)-\left(z-\lambda_{i}\right) I\right]^{-1} x \\ +> & =-\frac{1}{z-\lambda_{i}}\left[I-\left(z-\lambda_{i}\right)^{-1}\left(A-\lambda_{i} I\right)\right]^{-1} x. +> \end{aligned} +> ``` +> Now observe that $x \in \ker\left(A-\lambda_{i} I\right)$ implies +> ```math +> \left[I-\left(z-\lambda_{i}\right)^{-1}\left(A-\lambda_{i} I\right)\right] x +> = x - \left(z-\lambda_{i}\right)^{-1} 0 = x +> ``` +> and therefore that +> ```math +> R_{z}(A) x = -\frac{x}{z-\lambda_{i}}. +> ``` +> Based on this +>```math +>P_{i} \ x=\left[-\frac{1}{2 \pi i} \oint_{\contour_{i}} R_{z}(A) d z\right] x=\frac{x}{2 \pi i} \oint_{\contour_{i}} \frac{1}{z-\lambda_{i}} d z=x +>``` +>and $x \in \im \left(P_{i}\right)$. +> +>Finally, we show $\im \left(P_{i}\right) \subseteq \ker\left(A-\lambda_{i} I\right)$. +>With +> ```math +> \begin{aligned} +>\left(z-\lambda_{i}\right) R_{z}(A) +> &= (zI - A + A - \lambda_{i}I) R_{z}(A)\\ +> &=-I+\left(A-\lambda_{i} I\right) R_{z}(A) +> \end{aligned} +>``` +>we deduce +>```math +> \begin{aligned} +> -\frac{1}{2 \pi i} \oint_{\contour_{\lambda_{i}}}\left(z-\lambda_{i}\right) R_{z}(A) d z & =-\frac{1}{2 \pi i}\left(A-\lambda_{i} I\right) \oint_{\contour_{\lambda_i}} R_{z}(A) d z \\ +> & =\left(A-\lambda_{i} I\right) P_{i} +> \end{aligned} +>``` +>and more generally +>```math +> -\frac{1}{2 \pi i} \oint_{\contour_{\lambda_{i}}}\left(z-\lambda_{i}\right)^{k} R_{z}(A) d z=\left(A-\lambda_{i} I\right)^{k} P_{i} . \tag{1} +>``` +>Since the resolvent has no essential singularities, there exists a $k>0$ integer for which the left-hand side of (1) is zero. +>Since $\forall x \in \im \left(P_{i}\right)$, $x=P_{i} \ x$ this shows $x \in \ker\left(A-\lambda_{i} I\right)^{k}.$ +>For Hermitian, and thus diagonalisable matrices, +>```math +> \ker\left(A-\lambda_{i} I\right)^{k}=\ker\left(A-\lambda_{i} I\right) \quad \forall k \geq 1 +>``` +> thus $\im \left(P_{i}\right) \subseteq \ker \left(A-\lambda_{i} I\right)$. $\hspace{9cm} \square$ +""" + +# ╔═║ f0bbb6e5-5bbf-4c5b-895c-6d4531b75138 +md""" +!!! tip "Remark" + Making use of the eigenbasis $\{x_{i}\}_{i}$ of $A$ we can also denote + ```math + \indicator_{\{\lambda_{i}\}}(A) = + \sum_{i=1}^{n} \indicator_{\{\lambda_{i}\}} (\lambda_{i}) x_{i} x_{i}^{H} + =x_{i} x_i^H + = \ker (A-\lambda_{i} I )=P_{i} + ``` +This proves that the resolvent and associated spectral projectors contain all the information about the eigendecomposition of $A$. + +""" + +# ╔═║ cd7ed2fa-e9fa-45f9-ac6a-ba9a517e5593 +md""" +## Analyticity + +We proceed by considering the setting of our linear perturbation $A=A+t \Delta A$, which we study by considering the associated resolvent +```math +R_{z}(A(t))=(\tilde{A}+t \Delta A-z I)^{-1} +``` +First we note + +""" + +# ╔═║ 91ac9688-aa16-4d5c-8629-407141c9eee0 +md""" +!!! note "Proposition 3" + The resolvent $R_{z}(A(t))$ is analytic with respect to $t$ in the open disk of the complex plane with + ```math + |t|< \frac1{\spectralradius( R_{z}(\tilde{A}) \Delta A )} + ``` + +> *Proof.* +> Note that +> ```math +> \begin{aligned} +> R_{z}(A(t)) & =(\tilde{A}+t \Delta A-z I)^{-1} \\ +> & =R_{z}(\tilde{A})\left[I+t R_{z}(\tilde{A}) \Delta A\right]^{-1} +> \end{aligned} +> ``` +> which is analytic in $t$ as long as $\spectralradius (t R_{z}(\tilde{A}) \Delta A )<1$ +> $\hspace{6cm} \square$ + +""" + +# ╔═║ 4851946b-d584-495e-adc1-008e91555ec6 +md""" +!!! note "Theorem 4" + Let $\Gamma$ be a curve around one or a few eigenvalues of $\tilde{A}$ and let + ```math + \spectralradius_a = \inf_{z \in \Gamma}\left[\spectralradius (R_{z}(\tilde{A}) \Delta A )\right]^{-1} \text {. } + ``` + + Then $\spectralradius_{a}>0$ and the spectral projector + + ```math + P(t)=-\frac{1}{2 \pi i} \oint_{\Gamma} R_{z}(A(t)) d z + ``` + + is analytic in the disk $|t|<\spectralradius_a$. +""" + +# ╔═║ 163c66ac-6c48-476d-a4d2-9f44ef54dbe9 +md""" +> *Proof.* +> This result boils down to proving that under the given conditions the resolvent $R_{z}(A(t))$ is analytic for each $z \in \Gamma$, which is clearly the case if +> +> ```math +> |t|< \inf _{z \in \Gamma} \frac1{\spectralradius (R_{z}(\tilde A) \Delta A )} =\spectralradius_a . +> ``` +>We now show that $\spectralradius_{a}>0$. Consider +>```math +> \spectralradius (R_{z}(\tilde{A})\, \Delta A ) +> \leq \| \Delta A \, R_{z}(\tilde{A}) \| +> \leq\|\Delta A\| \ \|R_{z}(\tilde{A}) \|. +>``` +> Since $z \mapsto\left\|R_{z}(A)\right\|$ is continuous for $z \in \Gamma$, it reaches a maximum $z_{0}$ (extremal value theorem) for which +>```math +> \begin{aligned} +> \spectralradius (R_{z}(\tilde{A}) \, \Delta A) +> & \leq\|\Delta A\| \ \|R_{z}(\tilde{A}) \| +> \\ +> & \leq\|\Delta A\| \ \|R_{z_{0}}(\tilde{A}) \| \equiv K +> \end{aligned} +> ``` +> Hence, $1 / \spectralradius_{a} \leq K \iff \spectralradius_{a} \geq 1 / K>0$. +> $\hspace{8cm} \square$ +""" + +# ╔═║ 2d9827a3-2752-4bac-9c33-c46fa93b52f9 +md""" +An immediate corollary is that the **rank of $P(t)$ stays constant** as long as $|t|<\spectralradius_{a}$ : + +!!! note "Corollary 5" + The number $m$ of eigenvalues, counted with multiplicity, located in $\Gamma$ is constant given that $|t|<\spectralradius_a$. +""" + +# ╔═║ 2c1b0372-5232-45bf-95d1-1a6641508ef0 +md""" +!!! tip "Remark" + - Note, that $|t| < \spectralradius_a$ is usually too strong of a condition. + The real condition is that $P(t)$ is analytic. + - While individual eigenvalues may not be analytic is $t$, the average + ```math + \widehat{\lambda}(t)=\frac{1}{m} \sum_{i=1}^{m} \lambda_{i}(t) + ``` + of the $m$ eigenvalues of $A(t)$, which are in $\Gamma$, usually is analytic. + + - Another way to obtain $\widehat{\lambda}(t)$ is to consider the trace of the restriction of $A(t)$ to the invariant subspace $\im P(t)$, namely + ```math + \begin{align} + \operatorname{tr}\left(\left.A(t)\right|_{\im P(t)}\right) & =\operatorname{tr}\left[ (P(t) A(t) P(t)) \middle \vert_{\im P(t)}\right] \\ + & =\operatorname{tr}(P(t) A(t) P(t)) \\ + & =\operatorname{tr}(A(t) P(t)) \\ + & =\sum_{i=1}^{m} \lambda_{i}(t) \\ + & =m \widehat{\lambda}(t) + \end{align} + ``` +""" + +# ╔═║ 653598ee-c9d7-4731-9c8f-687d5962c0d6 +md""" +!!! note "Theorem 6" + The linear transformation $A(t) P(t)$ and its weighted trace $\widehat{\lambda}(t)$ are analytic in the disk $|t|<\spectralradius_{a}$. + +> *Proof.* +> $A(t) P(t)$ is analytic due to previous discussion. +> $t \mapsto \operatorname{tr}(M(t))$ is analytic if $M(t)$ is analytic, which is the case. +> $\hspace{11cm} \square$ + +""" + +# ╔═║ bd9f3ea5-4fd8-4155-bc80-5d658b304abe +md""" +Because of Theorem 6, a **simple eigenvalue** is **analytic** around $t=0$ and stays simple. +The vector $x_{i}(t)=P_{i}(t) \tilde{x}_{i}$ is an *eigenvector* of $A(t)$ associated to this simple eigenvalue if $\tilde{x}_{i}=x_{i}(t=0)$ is an **eigenvector** of $\tilde{A}=A(t=0)$. +Clearly $x_{i}(t)$ is **analytic, too** ! + +For a **multiple** eigenvalue the situation is more complex. +In general if an eigenvalue is of multiplicity $m$ then it will split into $m$ distinct small branches $\lambda_{i}(t)$. +Individually these are **not analytic** in $t$, but their average is. + +This makes it important to identify groups of eigenvalues in practice. +""" + +# ╔═║ c5067a9b-e202-4b03-9f54-aa367b2f4ebd +md""" +A final nice result is + +!!! note "Proposition 7" + Any eigenvalue $\lambda_{i}(t)$ of $A(t)$ inside the curve $\Gamma$ satisfies + ```math + \left|\lambda_{i}(t)-\tilde{\lambda}_{i}\right|= O(|t|) . + ``` + +> *Proof.* +> Due to Theorem 2, +> $\im (\tilde{P}_{i} ) = \ker (\tilde{A}-\tilde{\lambda}_{i} I )$ +> which implies $(\tilde A-\tilde{\lambda}_{i} I ) \tilde{P}_{i}=0$. +> +> Let us consider an eigenvector $x_i(t)$ with norm unity associated to $\lambda_{i}(t)$. We thus have +>```math +> \begin{align} +> \left(A(t)-\tilde{\lambda}_{i} I\right) P(t) \, x_i(t) & =\left(A(t)-\tilde{\lambda}_{i} I\right) x_i(t) \\ +> & =\left(\lambda(t)-\tilde{\lambda}_{i}\right) x_i(t) +> \end{align} +> ``` +> Taking norms on both sides : +>```math +>\begin{align} +> |\lambda(t)-\lambda_{i} | & = \| (A(t)-\tilde{\lambda}_{i} I ) P(t) \, x_i(t) \| \\ +>& \leq \| (A(t)-\tilde{\lambda}_{i} I ) P(t) \| \\ +>& = \| (A(t)-\tilde{\lambda}_{i} I ) P(t)- +> {\color{gray} \underbrace{\color{prooftext} (\tilde{A}- \tilde{\lambda}_{i} I ) \tilde{P}_{i} }_{=0} } \| \tag{2} \\ +>& =O(|t|) \tag{3} +>\end{align} +>``` +> where we went from (2) to (3) by considering that the right-hand side of (2) is analytic in $t$ and $\tilde A=A(0)$, $\tilde P_{i}= P(0)$, etc. +> $\hspace{10cm} \square$ + +""" + +# ╔═║ 22fe9a75-06bf-43b3-9cb0-92a9022ae653 +md""" +## Perturbation theory setting + +We return to the question of computing the lowest order variations of $\lambda(t)$ and $x(t)$ for $A(t)=\tilde{A}+t \Delta A$, having $(\tilde{\lambda}, \tilde{x})$ available. +We assume $\lambda(t)$ to be simple and isolated with $\Gamma$ a contour around $\tilde{\lambda}$. +Further we assume $|t|<\spectralradius_{a}=\inf _{z \in \Gamma} 1 / \spectralradius (R_{z} (\tilde A) \Delta A)$ such that $\lambda(t)$ and $x(t)$ are analytic. + +- Thus, we may expand them in a Taylor series: + ```math + \begin{align} + & \lambda(t)=\tilde{\lambda}+t \lambda^{\prime}(0)+\frac{1}{2} t^{2} \lambda^{\prime \prime}(0)+\cdots \\ + & x(t)=\tilde{x}+t x^{\prime}(0)+\frac{1}{2} t^{2} x^{\prime \prime}(0)+\cdots + \end{align} + ``` + +- Since $A(t) x(t)=\lambda(t) x(t)$ we have, to first order in $t$, + ```math + \begin{align} + \tilde{A} \tilde{x}+t\left(\Delta A \tilde{x}+\tilde{A} x^{\prime}(0)\right) + & =\tilde{\lambda} \tilde{x}+t\left(\lambda'(0) \tilde{x}+\tilde{\lambda} x^{\prime}(0)\right)+ O \left(| t |^{2}\right) + \\ + \Rightarrow \quad \Delta A \tilde{x}+\tilde{A} x^{\prime}(0) + & =\lambda^{\prime}(0) \tilde{x}+\tilde{\lambda} x^{\prime}(0)+O((t)) \tag{4} + \end{align} + ``` + +- Now we denote by $P_{\tilde \lambda}$ the projector into the eigenspace of $\tilde{\lambda}$ (generated by $R_{z}(\tilde{A})$ via a contour integral) and denote $Q_{\tilde \lambda}=I-P_{\tilde \lambda}$. +""" + +# ╔═║ eb10b66a-c103-4964-83e4-e980fdcaa36b +md""" +- Projecting (4) with $P_{\tilde \lambda}$ (by left multiplying with this operator), + we obtain after a rearrangement + ```math + \begin{align} + P_{\tilde{\lambda}} \, \Delta A \, \tilde{x}-\lambda^{\prime}(0) {\color{gray} \underbrace{\color{black} \, P_{\tilde{\lambda}} \, \tilde{x}}_{= \tilde x}} + = {\color{gray} \underbrace{\color{black} (\tilde{A}-\tilde{\lambda} I) \, P_{\tilde{\lambda}}} + _{\substack{= 0 \text{ since} \\ \ker (\tilde A - \tilde \lambda I) = \im P_{\tilde \lambda} }}} + x^{\prime}(0) +O(|t|) + \end{align} + ``` + Multiplying by $\tilde x^H$ from the left, we obtain + ```math + \lambda^{\prime}(0)=\tilde{x}^{H} \Delta A \, \tilde{x}=\langle\tilde{x}, \Delta A\, \tilde{x}\rangle + ``` + +- Projecting (4) with $Q_{\tilde \lambda}$, we obtain + ```math + Q_{\tilde{\lambda}} \, (\tilde{A}-\hat{\lambda} I)\, x^{\prime}(0)=-Q_{\tilde{\lambda}} \, \Delta A\, \tilde{x}+O(|t|) + ``` + Using the invariance of $Q_{\tilde \lambda}$ with respect to $\tilde A - \tilde \lambda I$, we get + ```math + \begin{align} + Q_{\tilde{\lambda}}(\tilde{A}-\tilde{\lambda} I)\, Q_{\tilde{\lambda}}\, x^{\prime}(0) &=-Q_{\tilde{\lambda}} \Delta A \, \tilde{x} +O(|t|) + \\ + \Rightarrow\quad Q_{\tilde{\lambda}}\, x^{\prime}(0) &=-Q_{\tilde{\lambda}}(\tilde{A}- \tilde{\lambda} I)^{-1} Q_{\tilde \lambda} \Delta A \, \tilde{x} \\ + \tag{$\ast$} + &=-R_{\tilde{\lambda}}(\tilde{A}) \, Q_{\tilde{\lambda}} \Delta A\, \tilde x + \end{align} + ``` + which defines the first-order change *orthogonal* to the eigenspace of $\tilde \lambda$. + +""" + +# ╔═║ 645a99c9-dc40-484d-bfe1-ea5a9745c009 +md""" +- To fix the remaining degree of freedom of $x^{\prime}(0)$, namely $\langle x^{\prime}(0), \tilde{x} \rangle$ we consider the normalisation of $x(t)$. + We require to first order + ```math + \begin{aligned} + 1 & =(x(t))^{H} x(t) \\ + & =\tilde{x}^H \tilde{x}+t\, \tilde{x}^{H} x^{\prime}(t)+t\,\left(x^{\prime}(t)\right)^{H} \tilde{x}+O\left(|t|^{2}\right) \\ + \Rightarrow 0 & =\operatorname{Re} \tilde{x}^{H} x^{\prime}(t)+O(|t|) + \end{aligned} + ``` + +- Additionally, we conventionally **choose** $\im \tilde{x}^H x^{\prime}(t)=0$. + This is **not necessary**, but reasonable to ensure $x(t)$ stays a real vector in case $\tilde{x}$ is real. + +- As a result $\tilde x$ and $x^{\prime}(t)$ are orthogonal and $P_{\tilde{Ξ»}}\, x'(0) = 0$ and ($\ast$) is the only contribution to $x'(t)$ in this convention. + +The final first-order changes are +```math +\boxed{ +\begin{array}{l} +\lambda'(0)=\langle\tilde{x}, \Delta A \tilde{x}) \\ +x^{\prime}(0)=-(\tilde{A}-\tilde{\lambda} I)^{-1} Q_{\tilde{\lambda}} \Delta A \tilde{x} +\end{array}} +\tag{PT} +``` + +""" + +# ╔═║ bc34fa94-f155-41fb-92d1-3e51db702f26 +md""" +## Interpretation and applications of this result + +If a modelling error / error is the computation causes us to employ $\tilde A$ instead of $A$ for the computation, then this result gives is the leading forward error with $\Delta A=A-\tilde{A}$. + +- We obtain the absolute forward error + ```math + \begin{align} + \|\lambda'(0) \| & \leq \|\Delta A\|\|x\|=\|\Delta A\| + \\ + \|x'(0) \| & \leq \|(\tilde{A}-\tilde{\lambda} I)^{-1} Q_{\tilde{\lambda}} \Delta A \| + \\ + & \leq \|(\tilde{A}-\tilde{\lambda} I)^{-1} Q_{\tilde{\lambda}} \|\| \Delta A\| + \\ + &= \frac{1}{\delta}\|\Delta A\| + \end{align} + ``` + where $\delta = \min _{\lambda \in \sigma(A)}|\lambda-\tilde{\lambda}|$ is the gap. +- Since the backward error is just $\| \Delta A\|$, this implies that the condition number is 1 for the eigenvalue computation and is $1 / \delta$ for the eigenvector computation. + +""" + +# ╔═║ f3fe5786-f3d9-4c14-9f51-911732a9dfb0 +md""" +#### Newton's method + +With respect to the exact eigenproblem $A(1)=\tilde{A}+\Delta A$, the term $\Delta A \, \tilde{x}$ is exactly the residual of $(\tilde \lambda, \tilde{x} )$. +We now consider one step of Newton's method for solving the system of equations: + +```math +\begin{align} +(A-\lambda I) x &= 0 \\ +\frac{1}{2} x^{H} x-1 &= 0 +\end{align} +``` + +with $A=A(1), \lambda=\lambda(1), x=x(1)$, where we desire the unknown eigenpair $(\lambda, x)$ while having the approximation $(\tilde \lambda, \tilde{x} )$. +One Newton step goes to +```math + \begin{pmatrix} + \tilde{x}_{\text {new}} \\ + \tilde{\lambda}_{\text {new}} + \end{pmatrix} + = + \begin{pmatrix} + \tilde{x} \\ + \tilde{\lambda} + \end{pmatrix} + - + \begin{pmatrix} + A-\lambda I & -\tilde{x} \\ + \tilde{x}^H & 0 + \end{pmatrix}^{-1} + \begin{pmatrix} + r \\ + 0 + \end{pmatrix} +``` +where $r=A\, \tilde{x} - \tilde{\lambda}\, \tilde{x}=\Delta A\, \tilde{x}$. +If we replace $A-\lambda I$ by the approximation $\tilde A-\tilde{\lambda} I$ we obtain a Quasi-Newton scheme, which under close inspection turns out to be exactly equivalent to (PT). + +Note that the normalisation of $x$ in the correction with respect to $\tilde{x}$ is again enforced by the constraint $\frac{1}{2} x^{H} x-1=0$. + +Typically, $\tilde{x}_{\text {new}}-\tilde{x} \perp \tilde{x}$ is manually enforced by inverting $Q_{\tilde \lambda}\, (\tilde{A}-\tilde{\lambda} I)\, Q_{\tilde \lambda}$ instead of $(\tilde A - \tilde \lambda I)$. +""" + +# ╔═║ 1a28c717-c969-4ce9-be55-80b566f6436d + + +# ╔═║ ed155363-2f4a-4aa8-9b41-79a5fa28c058 +TableOfContents() + +# ╔═║ e5b16293-ca61-4fc6-b89e-82100792efc7 +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 305) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 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"4f6342f7-b3d2-589e-9d20-edeb45f2b2bc" +version = "1.8.6+0" + +[[deps.Xorg_libXau_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8" +uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec" +version = "1.0.11+0" + +[[deps.Xorg_libXdmcp_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7" +uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05" +version = "1.1.4+0" + +[[deps.Xorg_libXext_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"] +git-tree-sha1 = "d2d1a5c49fae4ba39983f63de6afcbea47194e85" +uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" +version = "1.3.6+0" + +[[deps.Xorg_libXrender_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"] +git-tree-sha1 = "47e45cd78224c53109495b3e324df0c37bb61fbe" +uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" +version = "0.9.11+0" + +[[deps.Xorg_libpthread_stubs_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9" +uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" +version = "0.1.1+0" + +[[deps.Xorg_libxcb_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] +git-tree-sha1 = "bcd466676fef0878338c61e655629fa7bbc69d8e" +uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" +version = "1.17.0+0" + +[[deps.Xorg_xtrans_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77" +uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" +version = "1.5.0+0" + +[[deps.Zlib_jll]] +deps = ["Libdl"] +uuid = "83775a58-1f1d-513f-b197-d71354ab007a" +version = "1.2.13+1" + +[[deps.Zstd_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "555d1076590a6cc2fdee2ef1469451f872d8b41b" +uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" +version = "1.5.6+1" + +[[deps.libblastrampoline_jll]] +deps = ["Artifacts", "Libdl"] +uuid = 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\ No newline at end of file diff --git a/08_Hilbert_spaces.jl b/08_Hilbert_spaces.jl new file mode 100644 index 0000000..36fcf4e --- /dev/null +++ b/08_Hilbert_spaces.jl @@ -0,0 +1,1044 @@ +### A Pluto.jl notebook ### +# v0.20.3 + +using Markdown +using InteractiveUtils + +# ╔═║ 46704182-ef55-11ee-31c7-db589297db34 +begin + import TikzPictures.TikzPicture + using LaTeXStrings + using PlutoTeachingTools + using PlutoUI + using HypertextLiteral + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ 7d7ac845-c19c-4aff-84ed-8a486eb1bc56 +md""" +# Hilbert Spaces +""" + +# ╔═║ 49f2bf4a-8f1a-4122-aad6-8b50db93ac1f +md""" +In our introductory discussion abut quantum mechanics we already discussed the importance of the probability density $|\Psi(x)|^{2}$. +Clearly for this to make sense the wavefunction $\Psi: \mathbb{R}^{n} \rightarrow \mathbb{C}$ needs to be normalized, i.e. the integral +```math +\int \overline{\Psi(x)} \Psi(x) d x +``` +needs to be un-infinite. +Such an integral is *not* well-defined for ordinary functions $\mathbb{R}^{n} \rightarrow \mathbb{C}$ as not all functions are square-integrable over $\mathbb{R}^{n}$ (e.g. a non-zero constant is not). +This motivates the study of function spaces equipped with norms and the structure the norms impose. + +Typical norms for function spaces are the $L^p(\Omega)$-norms (or just $L^{p}$-norms when $\Omega \subset \mathbb{R}^{n}$ is clear from context) : +```math +\| f \|_{p} \equiv\left(\int_{\Omega}|f|^{p}\right)^{1 / p}. +``` +With this motivation in mind we will now study the structure of normed vector spaces and their companions such as Hilbert and Sobolev spaces. + +""" + +# ╔═║ 526efb20-8bfd-4060-9171-979c43b92435 +md""" + +## Completeness + +!!! note "Definition (Completeness)" + A normed vector space $(V , \| \cdot \|)$ is called **complete** if every Cauchy sequence of vectors in $V$ converges to on element in $V$. + +!!! tip "Remark (Cauchy sequence)" + Let us recall that a sequence $\left(x_{n}\right)_{n \in \mathbb{N}}$ of elements $x_{n} \in V$ is Cauchy if + ```math + \forall \varepsilon>0 \quad \exists M=M(\varepsilon) \in \mathbb{N}: \quad\left\|x_{n}-x_{m}\right\|<\varepsilon \quad \forall n, m \geq M(\varepsilon) . + ``` + What the definition says is that for such sequences there is a unique $x_{*} \in V$ with + ```math + \forall \varepsilon>0 \quad \exists N=N(\varepsilon) \in \mathbb{N}: \quad\left\|x_{n}-x_{*}\right\|<\varepsilon \quad \forall x_{n} \geq N(\varepsilon) . + ``` + +""" + +# ╔═║ 338c4b80-c710-4e8a-8dcd-e11ff464cbc8 +md""" +To make completeness as a concept more clear consider a counter-example. + +- Let's consider the rational numbers $\mathbb{Q}$ + (which can be seen as a vector space over the field $\mathbb{Q}$ itself) + and ask whether it is complete with respect to $| \cdot |$. + +- Consider the sequence + ```math + x_{n}=\sum_{k=0}^{n} \frac{1}{k !} \in \mathbb{Q} + ``` + of rational numbers. + It is well known that + ```math + \lim _{n \rightarrow \infty} x_{n}=e \notin \mathbb{Q} . + ``` + Thus, $\mathbb{Q}$ is not complete. + +- One may, however, build the **completion** of $\mathbb Q$ by including all possible limit points of all sequences with elements in $\mathbb Q$. This is one way to define the real numbers. + +A subtle point about completeness is that it depends on the norm which is used to determine whether a sequence is Cauchy or not. +In other words, a vector space may be complete with respect to one norm but not another. +Similarly, the completion of a space with respect to different norms may yield different spaces. + +In practice the choice of norm is only important for infinite-dimensional vector spaces as for finite-dimensional real/complex vector spaces all norms are equivalent. + +""" + +# ╔═║ 09efc623-3c15-4b91-8c3d-bda48baa8727 +md""" +!!! note "Definition (Banach space)" + A normed vector space which is complete is called a **Banach space**. + +!!! warning "Example 1 (Rⁿ)" + $\mathbb{R}^{n}$ with any $p$-norm $\|\cdot\|_{p}$. + +!!! warning "Example 2 (Lα΅–)" + For $1 \leq p<\infty$ the $L^{p}$ - spaces + ```math + L^{p}(\Omega)=\left\{ f: \Omega \rightarrow \mathbb{C} \ \middle | \ \int_{\Omega} |f(x)|^{p} d x<\infty \right\} + ``` + with $\Omega \subset \mathbb R^d$. + + +""" + +# ╔═║ c2b6dc90-b1ca-4641-9bfe-44d1558574f4 +md""" +!!! note "Definition (Hilbert space)" + A Banach space where the norm is induced by an inner product is a **Hilbert space**. + +!!! warning "Example 3 (Square integrable functions)" + An important Hilbert space is the space of square integrable functions + ```math + L^{2}\left(\mathbb{R}^{d}\right)=\left\{\psi: \mathbb{R}^{d} \rightarrow \mathbb{C} \ \middle | \ \int_{\mathbb{R}^{d}} | \psi(x) |^{2} d x<\infty\right\} + ``` + with inner product + ```math + \langle\psi, \varphi \rangle_{L^{2}}=\int_{\mathbb{R}^{d}} \overline{\psi(x)} \varphi(x) d x . + ``` +""" + +# ╔═║ 1100206d-b44d-4625-b425-f506cc258337 +md""" + +## Compactness + +We will now discuss compactness, a notion which we will meet in various places in the rest of the lecture. +Here we need it to introduce two important variants of $L^p$ spaces. + +!!! note "Definition (Compactness)" + Let $V$ be a normed vector space. + A subset $K$ of $V$ is called compact if every sequence $(x_{n} )_n \subset K$ has a converging subsequence whose limit is an element of $K$. + + +A colloquial way of stating this is : +!!! tip "" + On compact sets Bolzano-Weierstrass works. + +The extraction of subspaces is central in analysis. +For example the *extremal value theorem*, which we used to prove the existence of eigenvalues, is based on this argument and can be generalized to compact sets. + +!!! tip "Remark (Topological compactness)" + Equivalently to this definition in metric spaces is a coverage-based notion of compactness ($K \subseteq V$ is compact if every coverage of open sets admits a coverage using only a finite subcollection) due to the Heine-Borel theorem. + +!!! tip "Remark (Compactness in finite dimensions)" + In finite dimensions : + ```math + \text{Compact } \iff \text{ Closed and bounded} + ``` + +In infinite dimensions things are trickier as we will see below, pointing as to the fact that our eigenvalue existence proof (Lemma 2.2) will not go through for general operators. +""" + +# ╔═║ 4b412cc4-514c-4f30-b70d-babf8e58979f +md""" +!!! note "Theorem 1" + Let $V$ be a normed vector space for which every bounded sequence in $V$ contains a convergent subsequence. + Then, $\dim (V)<\infty$. + +> *Proof* by contradiction. +> Assume $\dim (V)=\infty$. We construct a bounded sequence that has no convergent subsequence: +> +> - Take $x_{1} \in V$ such that $\|x_{1} \|=1$ +> - Choose $x_{2} \in V$ such that $\|x_{2} \|=1$ and $\|x_{1}-x_{2}\| \geq 1 / 2$. +> - By the Riesz lemma we can choose $x_{k} \in V$ such that $\left\|x_{k}\right\|=1$ and $\left\|x_{k}-x_{i}\right\| \geq 1 / 2$ $\forall i=1, \dots, k-1 .$ +> +> This constructs a sequence $\left(x_{n}\right)_{n} \subset V$, such that $\left\|x_{n}\right\|=1$ (i.e. it is bounded), but $\left\|x_{n}-x_{m}\right\| \geqslant 1 \quad \forall m \neq n$. +> As the Cauchy criterion is never satisfied we cannot extract a convergent subsequence. +> $\hspace{11cm} \square$ + +""" + +# ╔═║ c14b2c6a-b7d7-465d-a591-474c703545d5 +md""" +## Weak derivatives & Sobolev spaces + +As was the case for matrices, computing Rayleigh quotients +```math +R_{\opH}(\psi)=\frac{\langle\psi, \opH \psi\rangle_{L^{2}}}{\langle\psi, \psi\rangle_{L^{2}}} \tag{1} +``` +for operators $\opH$ as well as the minimisation of these objects will be central when approximating spectra $\sigma(\opH)$. +We must therefore ensure that this quantity is well-defined. + +- All operators $\opH$ which we have studied so far and pretty much all Hamiltonians in quantum mechanics involve the Laplace operator (Kinetic energy). + However, if $\psi \in L^{2} (\mathbb{R}^{d} )$ there is no guarantee that derivatives of $\psi$ are still in $L^{2} (\mathbb{R}^{d} )$. + +- Thus, taking wavefunctions to be $L^{2}$ is *not* sufficient to ensure the $L^{2}$ inner product in (1) even makes sense. +- Therefore, we seek Hilbert spaces with a more restrictive structure, which ensures derivatives to be $L^{2}$ - integrable as well. + +Since we are only interested in integrating over derivatives, it turns out that we can allow ourselves a weaker notion of differentiability. +To define this, we first need to introduce two new function spaces : +- $C_{0}^{\infty}(\Omega)$ is the space of infinitely differentiable functions $\Omega \rightarrow \mathbb{C}$ with compact support. +- The space of locally integrable functions $L_\text{loc}^p$ : + +!!! note "Definition (Locally integrable functions)" + For $\Omega \subset \mathbb{R}^{d}$, we define + ```math + L_\text{loc}^{p}(\Omega)=\left \{ + f : \Omega \rightarrow \mathbb{C} \ \middle \vert \ + f\rvert_K \in L^{p}(K) \quad \forall K \subset \Omega, K \text { compact} \right \} + ``` + +We have $L^{p}(\Omega) \subset L_\text{loc}^{p}(\Omega)$, since local integrability is a less strict criterion : the growth of the function towards the boundary $\partial \Omega$ --- or towards $\infty$ --- is not limited. + +!!! tip "Remark" + Note $L^{p}(\Omega) \subset L_\text{loc}^{p}(\Omega) \subset L_\text{loc}^{1}(\Omega)$ for $p \geq 1$ and $\Omega \subset \mathbb{R}^{d}$ potentially unbounded, while $L^p(\Omega) \subset L^{1}(\Omega)$ only if $\Omega \subset \mathbb{R}^{d}$ is open and bounded. + +With this in mind, we can introduce the weak derivative : +""" + +# ╔═║ 374523ab-decb-42e9-88bf-3420f7552d39 +md""" +!!! note "Definition (Weak derivative)" + The function $u \in L_\text{loc}^{2}(\Omega)$ with $\Omega \subset \mathbb{R}^{d}$ open has a **weak derivative** along the coordinate $x_i$ in $L^{2}(\Omega)$ + if there exists a $g_{i} \in L^{2}(\Omega)$ with + + ```math + \begin{align} + \forall \varphi \in C_{0}^{\infty}(\Omega) + && + \left\langle g_{i}, \varphi \right\rangle_{L^{2}(\Omega)} + = + -\left\langle u, \frac{\partial \varphi}{\partial x_{i}}\right\rangle_{L^{2}(\Omega)}. + \end{align} + ``` + We then usually employ the same notation as for strong derivatives and e.g. denote $\frac{\partial u}{\partial x_{i}} = g_i$ **in the weak sense**. + + Defining for $\alpha= (\alpha_{1}, \ldots, \alpha_{n} ) \in \mathbb{N}_{0}^{n}$ the notation + ```math + \begin{align} + \|\alpha\|_{1}=\sum_{i=1}^{n} \alpha_{i} + && + D ^{\alpha}=\frac{\partial^{\|\alpha\|_{1}}}{\partial x_{1}^{\alpha_{1}} \ldots \partial x_{n}^{\alpha_{n}}} . + \end{align} + ``` + we analogously define higher weak derivatives $D^{\alpha} u \in L^{2}(\Omega)$ if there exists $g_{\alpha} \in L^{2}(\Omega)$ with + + ```math + \begin{align} + \forall \varphi \in C_{0}^{\infty}(\Omega) + && + \left\langle g_{\alpha}, \varphi \right\rangle_{L^{2}(\Omega)} + = + (-1)^{\|\alpha\|_{1}}\left\langle u, D^{\alpha} \varphi \right\rangle_{L^{2}(\Omega)} + \end{align} + ``` +""" + +# ╔═║ 972451e6-4e52-4d60-ac58-744e87d752d5 +md""" + +The motivation for this definition is partial integration, where the boundary terms vanish since $\varphi$ has compact support. +Explicitly, for $\Omega = (a,b)$, we have +```math + \int_a^b u(x) \frac{\partial \varphi(x)}{\partial x} dx = \underbrace{\bigg [ u(x) \varphi(x) \bigg ]^b_a}_{0 \text{ ($\varphi$ has compact support)}} - \int_a^b \frac{\partial u(x)}{\partial x} \varphi(x) dx +``` + + +If the strong derivative exists, the weak one agrees with it. +The weak derivative is well-defined. +""" + +# ╔═║ a84d9ce7-b0d9-4c88-a433-dcab67459a11 +md""" +This notion allows us to construct new Hilbert spaces : + +!!! note "Definition (Sobolev spaces)" + Let $\Omega \subset \mathbb{R}^{d} $ open. + The Sobolev space $H^n(\Omega), n \in \mathbb{N}$ + ```math + H^{n}(\Omega)= \{\psi \in L^{2}(\Omega) \mid D^{\alpha} \psi \in L^{2}(\Omega) \quad \forall \alpha,\|\alpha\|_{1} \leq n \} + ``` + with inner product + ```math + \langle f, g\rangle_{H^{n}}=\sum_{ \| \alpha \|_{1} \leq n}\left\langle D^{\alpha} f, D^{\alpha} g\right\rangle_{L^{2}} + ``` + and induced norm + ```math + \|f\|_{H^{n}}=\sum_{\| \alpha \|_{1} \leq n}\left\|D^{\alpha} f\right\|_{L^{2}} + ``` + is a *Hilbert space.* + + +!!! tip "Remark" + The Fourier transform is a useful tool to classify Sobolev spaces + ```math + f \in H^{n} (\mathbb{R}^{d} ) + \iff + \int_{\mathbb{R}^{d}}\left(1+|p|^{2 n}\right) |\hat{f}(p) |^{2} d p<\infty + ``` +""" + +# ╔═║ c0d4d23b-5b3e-441d-8161-59f8f78fbb52 +md""" +## Denseness and separability + +Before returning our focus to operators, the final concepts we need to discuss are denseness and separability. + +!!! note "Definition (Dense subspaces)" + A subspace $S \subset V$ is **dense** on $V$ if each vector $x_{*} \in V$ is either (1) also a member of $V$ or (2) one can find a Cauchy sequence $\left(x_{n}\right)_{n} \subset S$ which converges to $x_{*}$. + + +!!! tip "Remark" + $\mathbb{Q}$ is dense in $\mathbb{R}$. + +If $S \subset V$ is dense this means that, using only elements from $S$, we can construct an approximation to any $x_{*} \in V$ to any desired accuracy. +Clearly, finding good dense subspaces +is key to do numerical computations involving +infinite dimensional spaces like $H^{n}(\Omega)$. +In particular the ones with a countable number of basis functions are useful as this provides a natural way to start with a crude approximation (use few basis functions) and then keep refining until the desired accuracy is reached. + + +Fortunately, +""" + +# ╔═║ 04821b47-064b-407e-b6a6-aa8439b0f0f6 +md""" +!!! note "Definition (Separability)" + A Hilbert space is separable if it has a dense countable subset. + +A Hilbert space $\hilbert$ is separable if and only if it admits a countable orthonormal basis $\{\varphi_{\mu} \}_{\mu=1}^{\infty} \subset \hilbert.$ +With this each element $\psi \in \hilbert$ can be identified as +```math + \psi=\sum_{\mu=1}^{\infty} c_{\mu} \varphi_{\mu} +``` +to a square-summable infinite sequence $\left(c_{\mu}\right)_{\mu} \in \mathbb{C}$. +Every infinite-dimensional separable Hilbert space is thus isometrically isomorphic to the sequence space +```math + \ell^{2}(\mathbb{C})= + \left\{a: \mathbb N \rightarrow \mathbb{C} \ \middle | \ \sum_{n=1}^{\infty} | a(n) |^{2}<\infty\right\}. + +``` + +Separability is also closely related to being able to approximate the infinite-dimensional space numerically. +""" + +# ╔═║ a6aafce2-6039-499e-a281-59b0483d8c4b +md""" + +!!! warning "Example 4 (Examples of separable Hilbert spaces)" + Examples of separable Hilbert spaces : + - $L^p(\Omega)$ for $1 \leq p<\infty$ if $\Omega \subset \mathbb{R}^{d}$ is open. + - $H^{n}(\Omega)$ as subspaces of $L^{2}(\Omega)$ + +!!! warning "Example 5 (Classic counterexample)" + $L^{\infty}([0,1])$ is *not* separable. + Consider the family of characteristic functions $f_{t}= \indicator_{[0, t]}$ with $0 0$ there exists a $k \in \mathbb{N}$ such that $\left\|g_{k}-x\right\|<\alpha$. + Therefore, there is a $k \in \mathbb{N}$ for each $0$toc""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 305) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +TikzPictures = "37f6aa50-8035-52d0-81c2-5a1d08754b2d" + +[compat] +HypertextLiteral = "~0.9.5" +LaTeXStrings 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\ No newline at end of file diff --git a/09_Operators.jl b/09_Operators.jl new file mode 100644 index 0000000..992ff1e --- /dev/null +++ b/09_Operators.jl @@ -0,0 +1,1403 @@ +### A Pluto.jl notebook ### +# v0.19.43 + +using Markdown +using InteractiveUtils + +# ╔═║ e80f5b7f-dfa0-4785-86c9-fdc6c387f3ca +begin + using HypertextLiteral + using PlutoUI + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ 1f6efd53-2429-46e7-a579-fedd920250ff +md""" +# Operators +""" + +# ╔═║ 041e2cfd-0ec9-4766-98dd-7a90b2a4da84 +md""" +Previously we discussed the essential properties of function spaces, which can be seen as infinite-dimensional generalization of Euclidean space where functions play the role of vectors. + +We noted a number of important properties (compactness, completeness, separability), which in infinite dimensions are considerably harder to obtain or verify then in finite dimensions. + +Along these lines, linear operators, i.e. objects mapping functions to functions, are the natural generalization of matrices. +Conversely, we can consider matrices as operators between finite dimensional spaces. +Once again, these objects highlight some subtle differences between infinite and finite dimensional spaces. +""" + +# ╔═║ c3fdd2eb-f9c0-49cc-8c56-274971a1fc3b +md""" + +## Linear Operators + +!!! note "Definition (Operator)" + Let $V, W$ be $\mathbb{C}$ vector spaces. + A map $\opA : V \rightarrow W$ is called **linear operator** or just **operator** if + ```math + \forall x, y \in V \quad \forall \alpha, \beta \in \mathbb{C} \quad \opA (\alpha x+\beta y)=\alpha \opA (x)+\beta \opA (y) + ``` + + If $W=\mathbb C$ we can also call the operator a **linear functional**. +""" + +# ╔═║ d3ad8a0a-fa8a-4864-ada3-fe237e65c088 +md""" +!!! warning "Examples of operators" + - The zero operator $\opA : V \rightarrow W: x \mapsto 0_{W}$ + - The identity operator $\operatorname{id} : V \rightarrow V: x \mapsto x$ + - Matrices, e.g. + ```math + A=\left(\begin{array}{lll} + 1 & 2 & 3 \\ + 4 & 5 & 6 + \end{array}\right) + ``` + defines the operator $\opA : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ : + ```math + \left(\begin{array}{l} + x_{1} \\ + x_{2} \\ + x_{3} + \end{array}\right)=x \mapsto A x=\left(\begin{array}{l} + x_{1}+2 x_{2}+3 x_{3} \\ + 4 x_1 +5 x_{2}+6 x_{3} + \end{array}\right) + ``` + + - Differentiation: with $V=C^{1}([a, b])$ and $W=C^{0}([a, b])$, $\opA (f)=f^{\prime}$ is a linear operator. + + +""" + +# ╔═║ c5c84bdb-957d-4909-be69-e8bf31051fb6 +md""" +!!! tip "Remarks (Properties of matrices that keep holding)" + Zeros are mapped onto each other, since + ```math + 0_{W}=0_{\mathbb{C}} \opA(x)=\opA\left(0_{\mathbb{C}} x\right)=\opA \left(0_{V}\right) . + ``` + +!!! tip "" + For a subspace $\tilde{V} \leq V$ we have that $\opA (\tilde{V}) \leq W$ is a subspace of $W$ since, for all $\opA(x), \opA(y) \in \opA(\tilde{V})$, we have + ```math + \alpha \opA(x)+\beta \opA(y)= \opA \underbrace{(\alpha x+\beta y)}_{\in \tilde{V}} \in A(\tilde{V}) + ``` + which proves closure. + +!!! tip "" + By a similar argument one proves that linear operators conserve linear combinations + ```math + \opA \left(\sum_{i=1}^{n} \alpha_{i} x_{i}\right)=\sum_{i=1}^{n} \alpha_{i} \opA \left(x_{i}\right) + ``` + i.e. + ```math + \opA (\operatorname{span} \{x_{1},\dots,x_{n} \} )=\operatorname{span} ( \{\opA (x_{1} ), \dots, \opA(x_{n} ) \} ). + ``` + +!!! tip "" + Concatenations of two linear operators $\opA_{1}, \opA_{2}$ remain linear : + + ```math + \begin{align} + (\opA_{1} \circ \opA_{2} )(\alpha x+\beta y) & =\opA_{1} (\opA_{2}(\alpha x+\beta y) ) \\ + & = \opA_{1} (\alpha \opA_{2}(x)+\beta \opA_{2}(y) ) \\ + & =\alpha (\opA_1 \circ \opA_{2} )(x)+\beta (\opA_{1} \circ \opA_{2} )(y) + \end{align} + ``` + +!!! tip "" + Kernel and image are subspaces : + ```math + \begin{align} + \ker(A) &= \{x \in V: \opA (x)=0\} \leq V \\ + \im (A) &= \{y \in W: \exists x \in V \text{ s.t. } \opA (x)=y\} \leq W + \end{align} + ``` +""" + +# ╔═║ 8ce37b41-90c1-4a85-b6eb-814e0e51ca6f +md""" +!!! note "Definition (Operator norm and bounded operators)" + Let $\left(V,\|\cdot\|_{V}\right)$ and $\left(W,\|\cdot\|_{W}\right)$ denote two Banach spaces. + A linear operator $\opA: V \rightarrow W$ is called **bounded** if the norm + ```math + \|\opA \|_{\boundedoperators (V, W)} \coloneqq \sup _{0 \neq x \in V} \frac{\| \opA (v)\|_{W}}{\| v \|_{V}}<\infty + ``` + is bounded. + The set $\boundedoperators (V,W)$ of all bounded operators with this norm is a Banach space. + +""" + +# ╔═║ 9e4cdc94-8ffa-42de-b541-c8beddec8e50 +md""" +Similar to the case of completeness, the choice of norm is essential for boundedness. +To illustrate this, we consider the boundedness of the derivative operator. + +- Consider the operator $\partial: X \rightarrow Y$ mapping between the normed spaces $X=C^{1}\left([0,1], \| \cdot \|_{\infty}\right)$ and $Y=C^{0}\left([0,1],\|\cdot\|_{\infty}\right)$, i.e. where the supremum norm is used to measure distances. + The action of $\partial$ is $\partial f=f^{\prime}$ for $f \in X$. + In this case, $\partial$ is unbounded. + To see that consider the sequence $\left\{f_{n}(x)\right\}_{n=1}^{\infty} \subset X$ with $f_{n}(x)=\sin(2 \pi n x)$. + Clearly + ```math + \begin{align} + \sup _{0 \neq f \in C^{1}([0,1])} \frac{\|\partial f\|_{\infty}}{\|f\|_{\infty}} + & \geq \sup_{n=1, \ldots, \infty} \frac{\left\|\partial f_{n}\right\|_{\infty}}{\left\|f_{n}\right\|_{\infty}} + \\ + & = \sup_{n=1, \dots, \infty} \frac{2 \pi n}{1}=\infty, + \end{align} + ``` + which implies the unboundedness of $\partial$. + +- Now we consider $\partial: \tilde{X} \rightarrow Y$ with $\tilde{X}=C^{1}( [ 0,1], \| \cdot \|_{C^1})$ and $Y=C^{0}\left([0,1],\|\cdot\|_{\infty}\right)$, where $\tilde{X}$ is now equipped with the stronger norm + ```math + \|f\|_{C^{1}}=\|f\|_{\infty}+\left\|f^{\prime}\right\|_{\infty} . + ``` + Then $\partial$ is bounded, since + ```math + \begin{align} + \sup_{0 \neq f \in C^1([0,1])} \frac{\| \partial f \|_\infty}{\| f \|_{C^1}} + &= \sup_{0 \neq f \in C^1([0,1])} \frac{\| f' \| _\infty}{\| f \|_\infty + \| f' \|_\infty} + \\ + & \leq \sup _{0 \neq f \in \tilde{x}} \frac{\left\|f^{\prime}\right\|_{\infty}}{\left\|f^{\prime}\right\|_{\infty}}=1 . + \end{align} + ``` + +""" + +# ╔═║ 20da7e84-32e7-49d3-97a1-2e0793878ef5 +md""" +!!! note "Definition (Continuity)" + Let $\left(V,\|\cdot\|_{V}\right)$ and $\left(W,\|\cdot\|_{W}\right)$ be two Banach spaces. + An operator $\opA: V \rightarrow W$ is called **continuous** if for every sequence $\{x_{n} \}_{n} \subset V$ with $x_{n} \rightarrow x \in V$ as $n \rightarrow \infty$ it holds + ```math + \opA x_{n} \rightarrow \opA x \quad \text { as } \quad n \rightarrow \infty . + ``` + +!!! note "Theorem 1 (Boundedness-continuity equivalence)" + Let $\left(V,\| \cdot \|_{V}\right)$ and $\left(W,\|\cdot\|_{W}\right)$ be two Banach spaces. + A linear operator $\opA : V \rightarrow W$ is bounded if and only if it is continuous. +""" + +# ╔═║ 0816a190-9344-4a39-941b-5b9b375dc7b4 +md""" +> *Proof.* +> +> $\boxed \Rightarrow$ If $\opA$ is bounded then there exists a constant $c$ such that for all $x \in V$ we have $\| \opA x\left\|_{W} \leq c \right\| x \|_{V}$. Let $(x_{n})_n \subset V$ be a convergent sequence with limit $x \in V$. Then +> ```math +> \| \opA x_{n}- \opA x \|_{W} \leq c \|x_{n}-x \|_{V} +> ``` +> by linearity of $\opA$. +> Since $\left\|x_{n}-x\right\| \rightarrow 0$ as $n \rightarrow \infty$, $\opA x_{n} \to \opA x$ as $n \rightarrow \infty$, i.e. $\opA$ is continuous. +> +> +> $\boxed \Leftarrow$ +> If $\opA$ is continuous we can state an $\varepsilon$-$\delta$ criterion: For any $\varepsilon>0$, there is a $\delta>0$ such that +> ```math +> \| \opA x_{1}-\opA x_{2} \|_{W} \leq \varepsilon \quad \forall x_{1}, x_{2} \in V \text{ with } \|x_{1}-x_{2} \|_{V} \leq \delta +> ``` +> We set $\varepsilon=1$ and consider the case $x_{1}=0$ and $x_{2}=x \in V$ arbitrary. +> Then there exists a $\delta>0$ such that +> ```math +> \| \opA x \|_{W} \leq 1 \quad \text { if } \quad \|x\|_{V} \leq \delta +> ``` +> By linearity, we equivalently find, for all $\tilde{x} \in V$, +> ```math +> \|\tilde{x}\|_{V} \leq 1 \Rightarrow\| \opA \tilde{x}\|_{W} \leq 1 / \delta +> ``` +> Therefore, we can define a constant $C>0$ +> ```math +> C=\max_{\substack{ x \in V\\\| x \| \leq 1}} \frac{\| \opA x\|_{W}}{\|x\|_{V}} \leq 1 / \delta<\infty \tag{1} +> ``` +> Denoting $x=\|x\|_V \hat{x}$ with $\| \hat x \|=1$, for all $x \in V$ we have +> ```math +> \|\mathcal{A} x\|_{W}=\|x\|_{V}\| \opA \hat{x}\|_{W} \stackrel{(1)}{\leq}\|x\|_{V} C \|\hat{x}\|_{V}=C\|x\|_{V} +> ``` +> which proves that $\opA$ is bounded. +> $\hspace{10cm} \square$ +""" + +# ╔═║ d332ddd1-942a-41c0-b8d1-cef005693f0d +md""" +!!! tip "Remark (Boundedness in finite dimensions)" + In the finite dimensional case all linear operators (matrices) are bounded and thus continuous. + +!!! tip "Remark (Boundedness of SchrΓΆdinger operators)" + In passing we note that typical SchrΓΆdinger operators are not bounded. + +""" + +# ╔═║ a2e86a6f-88e7-4cbd-90c6-6d0e20af7869 +md""" +In the following we will consider the specific case of linear operators on a Hilbert space $\hilbert$, i.e. operators of the form $\opA: D(\opA) \rightarrow \hilbert$. The **domain** $D(\opA) \leq \hilbert$ is a dense subspace of $V$. Unless otherwise noted the standard choice of the domain is +```math + D(x)=\{f \in \hilbert \mid \opA f \in \hilbert \}, +``` +i.e. the largest possible subspace of $\hilbert$, such that applying the operator does not take us ot of $\hilbert$. Unless otherwise noted we will always employ this domain. + +- Other choices of the domain are well possible and sometimes useful. Moreover and perhaps surprisingly it has a decisive influence on the properties of an operator. For example it is related to the operator being self-adjoint or not, as we will discuss. + + +- To illustrate the point of allowing a domain to be smaller than or different from the Hilbert space, we consider the SchrΓΆdinger operator of an isolated system, which has the form $\opH=-\Delta+ V$, where $V$ is a potential. + The natural Hilbert space for this setting is $L^{2}(\mathbb{R}^{d})$. + Thus to ensure $\opH f \in L^{2} (\mathbb{R}^{d})$ $\forall f \in D(\opH)$, we need to take $D(\opH)=H^{2} (\mathbb{R}^{d} )$, one of the Sobolev spaces we introduced last time. + Clearly both $D(\opH)= \hilbert =L^{2}\left(\mathbb{R}^{d}\right)$ or $D(H)=\hilbert=H^{2}\left(\mathbb{R}^{d}\right)$ are *not* reasonable as in both cases the Laplacian reduces the regularity. + + +- For convenience and when the underlying Hilbert space is clear, we frequently denote the operator norm as + ```math + \| \opA \|_{op} \equiv \sup _{0 \neq f \in D(\opA)} \frac{\| \opA f\|_{\hilbert}}{\|f\|_{\hilbert}}, + ``` + i.e. the same setup as the matrix norm, but with the Hilbert space norm and domain inserted. +""" + +# ╔═║ 85df6f49-6935-45f0-9526-0b2b4075f3af +md""" +## Spectra of operators + +Having discussed the basic properties of operators, we now turn our attention towards their spectra. + +- Generalizing from matrices, an eigenpair $(\lambda, \varphi) \in \mathbb{C} \times \hilbert$ of the operator $\opA$ satisfies + ```math + \tag{$\ast$} + \begin{align} + \opA \varphi=\lambda \varphi & \Leftrightarrow(\opA-\lambda) \varphi=0 + \\ + &\Leftrightarrow \varphi \in \ker(\opA-\lambda) + \end{align} + ``` + where by $\opA-\lambda$ we understand the operator $\opA -\lambda \operatorname{id}_\hilbert$. + +- Under a slight abuse of notation we can write + $\ker(\opA - \lambda) = (\opA-Ξ»)^{-1} \{0\}$, + i.e. the solution set of applying the **resolvent** + $R_{z}(\opA)=(\opA-z)^{-1}$ to zero. + +- For eigenvalues $\opA-Ξ»$ is clearly not invertible as $(\ast)$ + shows that $\opA-Ξ»$ is not injective. However, let's assume for a second + it was and that a unique non-zero solution $Ο† = (\opA-Ξ»)^{-1} 0$ existed. + Then we had + ```math + 0 < \|Ο†\| = \left\|(\opA-\lambda)^{-1} 0\right\|. + ``` + This implies that $(\opA-Ξ»)^{-1}$ *cannot be bounded*, because if it was + than there existed a constant $C>0$ such that + ```math + \left\|(\opA-\lambda)^{-1} f \right\| \leq C\|f\| \qquad \forall f \in \hilbert + ``` + and in particular + ```math + \left\|(\opA-\lambda)^{-1} 0\right\| = C \, \|0\| = 0, + ``` + which is a contradiction to $Ο†$ being non-zero. + +As in the finite-dimensional case we first construct the *resolvent set*, which includes all the points that cannot be eigenvalues, i.e. the ones where the resolvent exists ($\opA-z$ can be inverted) and also the above aspect of a non-bounded $(\opA-Ξ»)^{-1}$ is excluded: +""" + +# ╔═║ 59c06a2d-d980-458f-bf7a-44bb4f4a8a80 +md""" +!!! note "Definition (Resolvent set)" + Let $\hilbert$ be a separable Hilbert space, $\opA : D(\opA) \rightarrow \hilbert$. The **resolvent set** is + ```math + \resolvent(\opA)=\{z \in \mathbb{C} \mid (\mathcal{A}-z) : D(\opA) \rightarrow \hilbert \text{ is invertible } {\color{noteblue} \underbrace{\color{black} \text{with bounded inverse}}_{\text{This is new in $\infty$-dimensions}}} \} + ``` + where bounded inverse means that the operator + ```math + (\opA -z)^{-1}: \hilbert \rightarrow D(\opA) \leq \hilbert + ``` + exists and there is a $C>0$ such that + ```math + \left\|(\opA-z)^{-1} x\right\| \leq C \|x\| \quad \forall x \in \hilbert . + ``` + For $z \in \resolvent(x)$ the **resolvent** $R_{z}(\opA)=(\opA-z)^{-1}$ exists and is bounded. + +""" + +# ╔═║ 8db7d0bc-ff6b-4387-8009-d83e74faaad0 +md""" +!!! tip "Remark (Eigenvalues are in the complement of ρ)" + By construction the set $\resolvent(A)$ contains all $z \in \mathbb{C}$ for which $(\opA-z) x=y$ admits a unique solution $x \in D(\opA)$ for a given $y \in \hilbert$. + Thus, for $(\opA-z) x=0$, only the trivial solution $x=0$ is possible. + To obtain eigenvalues we thus have to study the complement. As before +""" + +# ╔═║ 86e5f05a-6562-48c4-80f4-10c7cee0698e +md""" +!!! note "Definition (Spectrum)" + The **spectrum** is $\sigma (\opA) = \mathbb C \setminus \resolvent(\opA)$. +""" + +# ╔═║ 9823dc80-2adb-4e21-9588-fdd7dc1b3545 +md""" +From the definition of $\resolvent(\opA)$ there can be three reasons for a value $\lambda \in \mathbb{C}$ to be in $\sigma(\opA)$, respectively not in $\rho(\opA)$, namely + +1. $(\opA-z)$ not injective. + +2. $(\opA-z)$ not surjective. + +3. $(\opA-z)^{-1}$ not bounded. + +Recall that injectivity and surjectivity together make ($\opA - z$) invertible. + +For eigenpairs $(\lambda, \varphi) \in \mathbb{C} \times \hilbert$ one desirable property is that $(\lambda, C \varphi)$ for $0 \neq C \in \mathbb{C}$ is also an eigenpair. +Thus, we want $(A-z) \varphi=0$ to admit multiple solutions, implying $(\opA-z)$ to be not injective. +""" + +# ╔═║ 5827ce09-7199-4357-a5f4-8ec7ae396fca +md""" +!!! note "Definition (Eigenvalue, point spectrum, eigenspace)" + The set + ```math + \sigma_{p}(\opA)=\{z \in \mathbb{C} \mid \opA-z \text{ is not injective} \} + ``` + is the set of **eigenvalues** or **point spectrum** of $\opA$. + To each eigenvalue we can associate an **eigenspace** + ```math + \ker(\opA-\lambda)=\{\varphi \in D(A) \mid \opA \varphi=\lambda \varphi\} \neq\{0\} . + ``` + +!!! tip "Remark (Spectra in finite dimensions)" + For $\hilbert=\mathbb C^{N}$, 3. is always given, and 1. and 2. are equivalent because + ```math + \operatorname{dim}(\ker(\opA-\lambda))+\operatorname{dim}(\im (\opA-\lambda))=N . + ``` + Therefore $\sigma(\opA)=\sigma_p(\opA) \neq \varnothing$ (see Lemma 3 in [Matrix eigenproblems](https://teaching.matmat.org/error-control/03_Matrix_eigenproblems.html)). +""" + +# ╔═║ 6b9eb655-048a-42f6-b50c-d928071636bb +md""" +!!! tip "Remark (Behaviour in infinite dimensions)" + - In infinite dimensions injective operators can fail to be surjective. + To illustrate this, let $\hilbert=\ell^{2}(\mathbb{R})$ and define $\opA : \hilbert \to \hilbert$ by $\opA\left(\left(x_{n}\right)_{n}\right)=\left(0, x_{1}, x_{2}, \ldots\right)$ for $\left(x_{n}\right)_{n} \subset \mathbb{R}$ (i.e. we just insert a zero element at the front of the sequence). + $\opA$ is injective since $\opA \left(\left(x_{n}\right)_{n}\right)=0$ implies $\left(x_{n}\right)_{n}=0$, but is not surjective. + So $0 \in \sigma(A)$, even though it is *not* an eigenvalue. + + + - The reverse (surjective but not injective) can also be constructed. + + + - In infinite dimensions $\sigma(\opA)$ can be empty. + Consider $\op P_x f=-i \frac{\partial}{\partial x} f$ on $\hilbert=L^{2}(( 0,1) )$ with $D ( \op P_{x} )=\{f \in H^{2}((0,1)) \text{ with } f(0)=0 \}$. We define + ```math + \left(\op S_{z} f\right)(x)=i \int_{0}^{x} e^{i z(x-y)} f(y) d y + ``` + as an operator on $\hilbert$. + We compute $1=(\op P_x -z) \op S_{z}= \op S_{z}( \op P _x -z)$ for all $z \in \mathbb{C}$. + So $\op P_x-z$ is invertible. + Still $\op S_{z}$ is bounded because $\left\|\op S_{z} f\right\|_{L^{2}} \leq \|f\|_{L^{2}}$, so have $z \in \resolvent(\opA)$. + +To conclude we state a generalisation of an earlier statement about the series expansion of the resolvent. +""" + +# ╔═║ 07bf8bf2-d2b1-4ed8-b95c-f6c7d4cc6e45 +md""" +!!! note "Lemma 2 (optional)" + Let $\opA : D(\opA) \rightarrow \hilbert$ be a linear operator and $z \in \resolvent(\opA)$. + Then, $B_r(z) \subset \resolvent(A)$ with radius + ```math + r=\frac{1}{\|\left(\opA-z)^{-1} \|\right.} + ``` + and the resolvent can be expressed as the series + ```math + R_{t}(\opA)=(\opA-t)^{-1}=\sum_{n \geq 0}(t-z)^{n}(\opA-z)^{-(n+1)} \quad \forall t \in B_{r}(z) + ``` + Since, $\forall z \in \resolvent(A)$, this allows us to construct an open neighbourhood, which is also part of $\resolvent(A)$, we deduce that $\resolvent(\opA)$ is *open* and $\sigma(\opA)$ is *closed*. +""" + +# ╔═║ 12967e6f-5cda-4e16-95ab-2b6ae841d7bc +md""" +> *Proof* (same as the argument as Proposition 3 of [perturbation theory](https://teaching.matmat.org/error-control/07_Perturbation_theory.html)). +> Let $z \in \rho(x)$ and $t \in B_{r}(z) \subset \rho(\opA)$. Then +> ```math +> \begin{align} +> (\opA-t)^{-1} &= [(\opA-z)-(t-z)]^{-1} \\ +> &= {\color{gray} \underbrace{\color{prooftext} (\opA-z)^{-1}}_{\substack{\text{Inverse exists} \\ \text{because } t \in \resolvent(\opA)}}} +> \left [\operatorname{id}-(t-z)(\opA-z)^{-1}\right]^{-1} +> \end{align} +> \tag{2} +> ``` +> Since $|t-z|<\frac{1}{\left\|(\opA-z)^{-1}\right\|}$, we have $\left\|(t-z)(\opA-z)^{-1}\right\|<1$, which admits a Neumann expansion in the last term of (2), giving the desired result. +> $\hspace{5cm} \square$ + +""" + +# ╔═║ b2af25d8-e00d-4478-8ddc-ca809478b4b1 +md""" +## Self-adjoint operators + +As discussed in principle multiple choices for the operator domain $D(\opA)$ can sometimes be reasonable and thus seems to leave considerable ambiguity. + +- As the SchrΓΆdinger operator example shows, there is clearly some form of a maximal domain, since we clearly need $\varphi \in D(\opH)=H^{2} (\mathbb R^{d})$ to assure $\opH \varphi \in L^{2} (\mathbb{R}^{d} )$. + But what about choosing a smaller domain, e.g. $D(H)=C_{0}^{\infty} (\mathbb{R}^{d} )$ ? + + +- It turns out (detailed discussion in the appendix) that the choice of a sufficiently large domain $D(\opA)$ is required for the operator to be *closed*, which is a slightly weaker form of being continuous. + Being closed is necessary for being self-adjoint and moreover if $\opA$ is not closed, then $\sigma(A)=\mathbb{C}$, which is not useful for a physical model. + + +- Roughly speaking we therefore want $D(\opA)$ to be exactly large enough to get self-adjointness. + + +Without going into all such detail we will now sketch the definition of a self-adjoint operator and refer to the appendix for more details. +""" + +# ╔═║ 9dd77c43-3d86-4ab0-9ca3-6389a32ea739 +md""" + +We first state the definition of the (Hilbert-)**adjoint operator**. + +!!! note "Definition (Adjoint operators)" + Given an operator $\opA: D(\opA) \rightarrow \hilbert$ + with $D(\opA) \subset \hilbert$, the adjoint operator + $\opA ^*: D (\opA^{*} ) \rightarrow \hilbert$ is defined by $y \mapsto \opA^{*} y$, + where $\opA^{*} y \in \hilbert$ is the unique element satisfying + ```math + \left\langle x, \opA^{*} y\right\rangle=\langle\opA x, y\rangle. \qquad \forall x \in D(\opA), \forall y \in D\left(\opA^{*}\right). + ``` + Notably in this definition $D (\opA^{*} )$ is taken to be the + *largest domain possible* for this equality to make sense. + Uniqueness of $\opA^{*} y$ given $y$ is than guaranteed + by the Riesz representation theorem. + +A slightly less strict definition is: + +!!! note "Definition (Symmetric operators)" + The operator $\opA$ on $\hilbert$ is called **symmetric** if, for all $x, y \in D(\opA) \times D(\opA)$, + ```math + \langle\opA x, y\rangle= \langle x, \opA y\rangle. + ``` + +This is a first step towards self-adjoint operators and typically what is called "Hermitian operator" in physics texts. + +Comparing this definition of symmetric operators and the adjoint definition above we notice the difference in the sets over which $y$ pans out. +Crucially, *symmetry and self-adjointness are not equivalent concepts*. +However, + +""" + +# ╔═║ 25804d6a-9eb4-4c24-89cf-96042eac70d8 +md""" +!!! note "Definition (Self-adjointness)" + An operator is self-adjoint if it is symmetric with $D(\opA^*)=D(\opA)$, or simply an operator for which $\opA=\opA^{*}$. + +The choice of domain is crucial for self-adjointness. +One can show that choosing both a smaller or a larger domain $D(\opA)$ destroys the self-adjointness of $\opA$. +""" + +# ╔═║ 8e9a3665-775b-4097-bb00-b6a337176cab +md""" +We now illustrate this concept by providing some examples of self-adjoint operators on $\hilbert=L^{2}(\mathbb R^{d})$. +We will note that the previously introduced Sobolev spaces arise naturally on the correct domain to ensure self-adjointness of physical operators. + +!!! warning "Laplace operator" + $\op A=-\laplacian$ with $D(-\Delta)=H^{2}(\mathbb{R}^{d})$ is self-adjoint, but with $D(-\Delta)=C_{0}^{\infty} (\mathbb R^{d})$ it is not. + +!!! warning "Momentum operator" + $\op P_{j}=-i \frac{\partial}{\partial x_{j}}$ with + $D(\op P_{j})=H^{2}(\mathbb{R}^{d})$ is self-adjoint. + +!!! warning "SchrΓΆdinger operators" + The SchrΓΆdinger operators $\opH=-\Delta+V$ in $\hilbert=L^{2}\left(\mathbb{R}^{3}\right)$ are self-adjoint for $V \in L^{2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$ and $D(\opH)=H^{2}(\mathbb R^{3})$. The condition on the potential is to be understood as $V=V_{S}+V_{L}$ with $V_{S} \in L^{2}(\mathbb{R}^{3})$ (short-range component) and $V_{L} \in L^{\infty}(\mathbb{R}^{3})$ (long-range component). + This is satisfied for + - Any bounded potential $V= \color{warnyellow} \underbrace{\color{black} 0}_{\in L^2(\mathbb R^3)} {\color{black}+} \underbrace{\color{black} V}_{\in L^\infty(\mathbb R^3)}$ + - The Coulomb potential + ```math + V=\frac{1}{|x|} \quad \text { with } \quad V_{S}= + {\color{warnyellow} \underbrace{\color{black} \frac{e^{-|x|}}{|x|}}_{\in L^{2}(\mathbb R^{3})} } + \text{ and } V_{L} = + {\color{warnyellow} \underbrace{\color{black} \frac{1-e^{-|x|}}{|x|}}_{\in L^\infty(\mathbb R^3)}} + ``` + +""" + +# ╔═║ 5f500170-8d73-4d2b-87cb-c706c4966ba5 +md""" +We arrive at the following statement, which illustrates once again the difference between finite and infinite dimensions : + +!!! note "Proposition 3" + If a symmetric operator $\opA : \hilbert \to \hilbert$ is defined on the entire $\hilbert = D(\opA)$, then $\opA$ is self-adjoint and bounded. +""" + +# ╔═║ 392444bd-c154-4088-aa00-2b32018a90d6 +md""" +## Summary + +- For infinite dimensions $D(\opA)=\hilbert$ in rarely feasible. + Choosing $D(\opA)$ sensibly (e.g. Sobolev spaces) is crucial to get self-adjointness and, as we will see, real and physically meaningful spectra. + + +- Many the special properties we discussed are automatically given in finite dimensions or can be easily checked (e.g. symmetry). + + +- We have the following implication relationships : +```math +\boxed{ +\begin{align} + \text{self-adjoint} &\Rightarrow \text{closed} + \\ + & \Rightarrow \text{symmetric} +\end{align}} +``` + +```math +\boxed{ +\begin{align} + \text{(bounded} &\Leftrightarrow \text{continuous}) \Rightarrow \text{closed} + \\ + &\Uparrow + \\ + \dim(\hilbert) &< \infty \Rightarrow {\color{gray} \underbrace{\color{black} \sigma(\opA) = \sigma_p(\opA)}_{\text{spectrum only has eigenvalues}}} + \\ + &\Downarrow + \\ + D(\opA) &= \hilbert \Rightarrow (\text{symmetric} \Leftrightarrow \text{self-adjoint}) +\end{align}} +``` + +""" + +# ╔═║ 23d46169-107b-41a7-8ef1-410f63c046bc +md""" +## Appendix : In-depth treatment of self-adjointness + +To arrive at the definition of self-adjoint operators we will first look at the graph of operators, which might seem trivial aud unrelated, but is closely related to the question of choosing a good domain $D(\opA)$ for an operator, and the relation between $D(\opA)$, the spectrum $\sigma (\opA)$, and self-adjointness. + +""" + +# ╔═║ d6cd60e6-a1d0-402b-98e2-7cb75588ea0e +md""" +### Graph and closure + +!!! note "Definition (Graph)" + Let $A: D(\opA) \rightarrow \hilbert$. The **graph** of the operator $\opA$ is + ```math + \graph(\opA)=\{(\varphi, \opA \varphi) \mid \varphi \in D(\opA )\} \subset D(\opA) \times \hilbert + ``` + +Not all subsets of $\hilbert \times \hilbert$ are graphs of operators as the following lemma shows. + +!!! note "Lemma 4" + A set $\graph \subset \hilbert \times \hilbert$ is the graph of an operator if and only if + + 1. $\graph$ is a vector subspace of $\hilbert \times \hilbert$ + + 2. $(0, y) \in \graph$ implies $y=0$ + + 3. The projection $D =\{x \in \hilbert \mid \exists y \in \hilbert$ with $(x, y) \in \graph\}$ is dense in $\hilbert$. +""" + +# ╔═║ 5f3954d3-bbe5-474a-834c-42e8603ff4bf +md""" +> *Proof*. +> +> $\boxed{\Rightarrow}$ : (1) - (3) follow directly since $\opA$ is a linear operator and $D(\opA)$ is dense in $\hilbert$. +> +> $\boxed{\Leftarrow}$ : (2) ensures that each $x$ is assigned a unique image $y = \opA x$, since due to (1) $\graph$ is a vector space. +> Therefore, the application of $\opA$ is well-defined and linear. +> (3) finally ensures the density of $D(\opA)$ in $\hilbert$. +""" + +# ╔═║ a41650f2-2e4d-4c76-abe7-65e8867065e5 +md""" +!!! note "Definition (Extension and restriction)" + Let $\opA$ and $\op B$ be two operators on $\hilbert$. + $\op B$ is called on **extension** of $\opA$ if $D(\opA) \subset D(\op B)$ and if $\forall x \in D(\op A): \op A x= \op B x$. + In that case we also call $\opA$ a **restriction** of $\op B$. + +!!! warning "Example" + One can define the derivative operator $f \mapsto f^{\prime}$ on $\hilbert =L^{2}(\mathbb{R})$. The domain $D(\op B)=C_{0}^{\infty}(\mathbb{R})$ (the space of compact supported smooth functions) is a valid choice since $C_{0}^{\infty} \subset L^{2}(\mathbb{R})$ leading to an operator $\opA$. Alternatively we could define $\op B$ with domain $D(\op B)= H^{1}(\mathbb{R})$, which is the largest possible domain for this operator. Since $C_{0}^{\infty} \subset H^1(\mathbb{R})$, $\op B$ is an extension of $\opA$ and $\opA$ is a restriction of $B$. +""" + +# ╔═║ 3c24ac4f-6903-4cfd-93f3-be7af58f1448 +md""" +!!! note "Definition (Closed operators)" + An operator $\opA$ with domain $D(\op A)$ is called **closed** if its graph $\graph(\opA)$ is closed in $\hilbert \times \hilbert$. + That is, if for each converging sequence $(x_{n} ) \subset D(\opA)$ with $x_{n} \rightarrow x$ and $\opA x_n \to y$, it holds that $x \in D(\opA)$ and $\opA x= y$. + + +!!! tip "Remark" + Being closed can be seen as a slightly weaker form of being continuous. + The latter requires that every converging sequence $x_{n} \to x$ *always* leads to a converging sequence $\opA x_{n} \to \opA x$. + For a closed operator, not all sequences $\left(\opA x_{n}\right)_{n}$ need to converge, and only for those that do we have the requirement $\opA x_{n} \rightarrow \opA x$. + + A consequence is + ```math + \begin{align} + \textsf{bounded / continuous} \Rightarrow \textsf{closed} + \end{align} + ``` + +""" + +# ╔═║ 3b36fc40-9c77-49a1-bc9a-abc5752a0daf +md""" +!!! note "Proposition 5" + If $\opA$ with domain $D(\opA)$ is not closed then $\sigma(\opA)=\mathbb{C}$. + +> *Proof* by contradiction. +> We assume $\sigma(\opA) \neq \mathbb{C}$ and show that $\opA$ is necessarily closed. +> In this case, there exists a complex $z \notin \sigma(\opA)$, and we consider a sequence $(x_{n}) \subset D(\opA)$ such that $x_{n} \rightarrow x$ and $\op A x_{n} \rightarrow y$. +> Since $\opA-z$ is invertible with bounded (and thus continuous) inverse we have +>```math +>\begin{aligned} +> (\opA-z)^{-1} \opA x_{n} & =(\opA-z)^{-1}(\opA-z+z) x_{n} \\ +> & =x_{n}+z(\opA-z)^{-1} x_{n} +>\end{aligned} +>``` +>Going to the limit we obtain +>```math +> (\opA-z)^{-1} y=x+z(\opA-z)^{-1} x +>``` +> making use of $(\opA-z)^{-1}$ being continuous. +> Therefore $x=(\opA-z)^{-1}(y-z x)$, which is in $D(\opA)$ since $(\opA-z)^{-1}$ maps $\hilbert \rightarrow D(\opA)$. +> Self-multiplying by $(\opA-z)$ we obtain +>```math +> (\opA x-z) x=y-z x \iff \opA x=y +> ``` +> This contradicts our original statement. $\hspace{8cm} \square$ +""" + +# ╔═║ 353bee69-8a2e-4da2-83b6-6a2f30621fd6 +md""" +Clearly, this result is in stark contrast with physical intuition, where we seek operators with real spectra, ideally on isolated countable points. + +We now illustrate the point that non-closed operators can arise if $D(\opA)$ is chosen too small. + + +- We consider the momentum operator on $\hilbert=L^{2}(\mathbb{R})$, i.e. $\op P : f \mapsto -i f^{\prime}$, and set $D(\op P)=C_{0}^{\infty}(\mathbb{R}).$ + We claim $\op P$ is not closed. + To see this consider the Gaussian $f=e^{-x^2} \in H^{1}(\mathbb{R}) \setminus C_{0}^{\infty}(\mathbb{R})$. + By density of $C_{0}^{\infty}(\mathbb{R})$ in $H^{2}(\mathbb{R})$ we can construct a sequence $(f_{n})_n \subset C_{0}^{\infty}(\mathbb{R})$ such that $f_{n} \rightarrow f$ and $f_{n}^{\prime} \rightarrow f^{\prime}$. + While $(f_{n},-i f_{n}') \in \graph(\op P)$, the limiting pair $\left(f, -i f^{\prime}\right)$ is not. + Thus $\op P$ is not closed. + + +- A natural idea is to attempt the "closure" of on operator by adding elements to $\graph(\opA)$ until it is a closed set. + + +- Since the set should still be the graph of an operator we need to ensure that the properties of Lemma 4 hold. (1) and (3) are easily kept in such a closure process. + However, (2) is less easily conserved. + + +- In particular, there are operators which cannot be closed. + Due to proposition 5 they are in general not of physical interest, and we can disregard such peculiarities. +""" + +# ╔═║ b0072ed0-6d00-457e-94e7-f6a400125d23 +md""" +!!! note "Definition (Closable operators)" + Let $\opA$ be on operator with domain $D(\opA)$. + One calls $\opA$ **closable** if it admits at least one closed extension. + In that case the closed extension with the smallest domain $D(\bar \opA)$ is denoted $\bar \opA$ and called the **closure** of $\opA$. +""" + +# ╔═║ c2a9e93c-e3ed-4040-a703-5bddedf50cb6 +md""" +!!! note "Theorem 6 (Closure of βˆ‚ and Ξ” in Rᡈ)" + In $\hilbert =L^{2} (\mathbb{R}^{d})$, let $\op P_{j}^\text{min}$ be the momentum operator defined by $\op P_{j}^\text{min} f=-i \partial_{x_j} f$ with domain $D(\op P_{j}^\text{min})=C_{0}^{\infty} (\mathbb R^{d})$ for $j=1, \ldots d$. + $\op P_{j}^{\text{min}}$ is closable with closure $\op P_{j} \equiv \overline{\op P_{j}^\text{min}}$ with domain $D(\op P_{j})=H^{1}(\mathbb{R}^{d})$. + + Let further $\op{L}^\text{min} f=-\laplacian f$ in the domain $D(\op L^\text{min})=C_{0}^{\infty} (\mathbb R^d)$, $\op L^\text{min}$ is a again closable with closure $\op L \equiv \overline{\op L^\text{min}}$ of domain $D(\op {L})=H^{2}(\mathbb{R}^{d} )$. +""" + +# ╔═║ d2b6c0c1-aaf8-439c-a02c-707079f9ece5 +md""" + +> *Proof.* +> We start by proving that $\op P_{j}$ is closed. +> Let thus $\left(f_{n}, \op P_{j} f_{n}\right)=\left(f_{n},-i \partial_{x_{j}} f_{n}\right)$ be a sequence of points on the graph of $\op P_{j}$, which converges to $(f, g)$ in $\hilbert \times \hilbert$. +> If we show $f \in D (\op P_j )$ and $g= \op P_{j} f$, then $\op P_{j}$ is indeed closed. +> For this, we use integration by parts against a test function $\varphi \in C_{0}^{\infty} (\mathbb{R}^{d} )$ and convergence in $L^{2} (\mathbb{R}^{d} )$ : +>```math +>\begin{align} +>i \int_{\mathbb{R}^{d}} \varphi(x) g(x) d x +>& =\lim _{x \rightarrow \infty} \int_{\mathbb{R}^{d}} \varphi(x) \partial_{x_{j}} f_{n}(x) d x +> \\ +>& =-\lim _{n \rightarrow \infty} \int_{\mathbb{R}^{d}} \partial_{x_{j}} \varphi(x) f_{n}(x) d x +> \\ +>& =- \int_{\mathbb{R}^{d}} \partial_{x_{j}} \varphi(x) f(x) d x . +>\end{align} +>``` +>This shows that $i g =\partial_{x_j} f$ in the sense of distributions and in particular (since $g \in L^{2}\left(\mathbb{R}^{d}\right)$) that $\partial_{x_j} f \in L^{2} (\mathbb{R}^{d} )$. +>Therefore $f \in H^{1} (\mathbb{R}^{d} )=D\left(\op P_{j}\right)$ and $g=\op P_{j} f_{n}$ which shows that $\op P_{j}$ is closed. +>Thus, due to $\op P_{j}^\text{min} \subset \op P_{j}$, $\op P_{j}^\text{min}$ is closable. +>An immediate consequence is that $\overline{\graph(\op P_j^\text{min})} \subset \graph(\op P_j)$. +> +> To show $\overline{\op P_j^\text{min}} = \op P_j$ it remains to show $\graph (\op P_j) \subset \overline{\graph(\op P_j^\text{min})}$. +> For this we rely on the denseness of $C_{0}^{\infty} (\mathbb{R}^{2} )$ in $D (\op P_{j})=H^{1}(\mathbb{R}^{d})$ and in $L^{2}\left(\mathbb{R}^{d}\right)$ : this ensures that for each $(f, -\partial_{x_{j}} f) \in L^{2}(\mathbb{R}^{d}) \times L^{2}(\mathbb{R}^{d})$ one can find a sequence $(f_{n}) \subset C_{0}^{\infty}(\mathbb{R}^{d})$ such that $f_{n} \rightarrow f$ and $\partial_{x_{j}} f_{n} \rightarrow \partial_{x_j} f$ strongly in $L^{2}(\mathbb{R}^{d})$. +> Thus $\graph (\op P_{j}) \subset \overline{\graph (\op P_{j}^\text{min})}$ and $\overline{\op P_{j}^\text{min}}=\op P_{j}$. +> +> The arguments for $\overline{\op L^\text{min}} = \op L$ are similar, relying again on the denseness of $C_0^\infty(\mathbb R^d)$ in $H^2(\mathbb R^d)$. +> $\hspace{13cm} \square$ +""" + +# ╔═║ 8ed5e1b6-a7be-4b59-9921-238a3212cad7 +md""" +!!! tip "Remark" + We notice the natural appearance of Sobolev spaces as the domains of standard operators of quantum mechanics in $L^{2} (\mathbb{R}^{d} )$. +""" + +# ╔═║ d0e94968-d053-454c-9a50-6d1097c264a8 +md""" +### Adjoint operators and graph + +Looking at the graph of an operator $\opA$ also allows for a different point of view on adjoints. + +- For the adjoint $\opA^*$ we would like to have + ```math + \langle y, \op A x\rangle=\left\langle \op A^{*} y, x\right\rangle \qquad \forall x \in D(\opA), y \in D(\opA^*) \tag{3} + ``` + + +- The idea is to define $\opA^*$ with the largest $D\left(\op A^{*}\right)$ possible to satisfy the above equality. + + +- Introducing the inner product + ```math + \langle (x_{1}, x_{2} ),(y_{1}, y_{2} ) \rangle_{\hilbert \times \hilbert}= \langle x_{1}, y_{1} \rangle + \langle x_{2}, y_{2} \rangle + ``` + on the space $\hilbert \times \hilbert$, Equation (3) can also be written as + ```math + \langle (y, \opA^{*} y ), (\opA x, -x ) \rangle_{\hilbert \times \hilbert}=0, + ``` + which suggests defining the graph of $\opA^*$ as the orthogonal complement of the flipped graph of $\graph (\opA)$ + ```math + \graph (\opA^{*} )= \{ (\opA x, -x ) \mid x \in D(\opA) \}^{\perp} + ``` + where by $\perp$ we consider orthogonality with respect to $\langle \cdot, \cdot \rangle_{\hilbert \times \hilbert}$. +""" + +# ╔═║ df3f76f1-116d-4863-8d6a-c32076c01d6b +md""" +- Again we use Lemma 4 to test whether this set is indeed the graph of an operator $\opA^*$ with domain $D(\opA^*)$. + + (i) is satisfied since the orthogonal complement of a set is always a vector space, thus $\graph (\opA^*)$ is necessarily a vector space. + + For (ii) we employ the denseness of $D(\opA)$ in $\hilbert$. + Note that $(0, y) \in \graph(\opA^{*} )$ if and only if $\langle x, y\rangle=0$ for all $x \in D(\opA)$ by definition. + Therefore consider + ```math + D(\opA)^{\perp}= \{y \in \hilbert \mid (0, y) \in \graph (\opA^{*} ) \} . + ``` + Since $D(\opA)$ is dense, this implies $D(\opA)^\perp = \{ 0 \}$ and in turn $y=0$ as required. + + The tricky part is (iii) : the denseness of $D (\opA ^{*} )$. + Consider $D (\opA^{*} ) ^\perp$. + We have $y \in D (\opA^{*} ) ^\perp$ if and only if + ```math + \begin{align} + {\color{gray} + \underbrace{\color{black} (y, 0) \in \graph (\opA^{*} )^{\perp}} + _{\substack{ + \text{Because Eq. (3) is exactly \emph{not} satisfied} \\ + \text{for those $x$, i.e. we are $\hilbert \times \hilbert$} \\ + \text{orthogonal to $\graph(\opA^*)$} + }} + } + &= ( \{ (\opA x, -x ) \mid x \in D(\opA) \}^{\perp} )^{\perp} + \\ + &=\overline{ \{ (\opA x, -x \mid x \in D(\opA) \} }. + \end{align} + ``` + or equivalently if $(0,y) \in \overline{\graph(\opA)}$. In other words we have found that + ```math + D (\opA^{*} )^{\perp}=\{x \in \hilbert \mid (0, x) \in \overline{\graph(\opA)}\} . + ``` + That is to say $D (\opA^{*} )$ is dense if and only if $\overline{\graph(\opA)}$ satisfies property (ii) of Lemma 4., i.e. if $\opA$ is closable. +""" + +# ╔═║ 1d249ea8-9b1a-4a47-bb08-9fdb2525ad3c +md""" +To summarise the discussion: + +- The adjoint $\opA^{*}$ with dense domain $D (\opA^{*} )$ is only well-defined if $\opA$ is closable. +- $\opA^{*}$ is always closed. + +The next result is no surprise after this discussion: + +!!! note "Lemma 7" + Let $\opA: D(\opA) \to \hilbert$ be a closable operator. + Then it holds that + ```math + \left ( \opA^* \right )^* = \overline{\opA} + ``` + and + ```math + \left ( \left ( \opA ^* \right )^* \right )^* = \opA^* = \overline{\opA^*}. + ``` +""" + +# ╔═║ a8ad5b99-db46-423f-8541-755c4a3099c8 +md""" +### Symmetry, graphs, and related extensions + +- Recall an operator $\opA$ defined on $D(\opA)$ is called **symmetric** if for all $x, y \in D(\opA)$ we have $\langle x, \opA y\rangle=\langle \opA x, y\rangle$. + +- Equivalently one could state that $\opA^*$ needs to be an extension of $\opA$, i.e. $\opA \subset \opA^*$ or yet equivalently $\graph(\opA) \subset \graph (\opA^{*} )$. + +- Since $\opA^{*}$ is always closed, $\opA \subset \opA^{*}$ implies that a symmetric $\opA$ is always closable with $\overline \opA= (\opA ^{*} )^{*}$. For a symmetric $\opA$ the operators $\opA ^{*}$ and $(\opA ^{*} )^{*}$ are thus two important closed extensions it always possesses. + +- If $\op A \subset \op B$ with $\op B$ a symmetric extension of $\opA$, then + ```math + A \subset \overline{\opA} \subset \op B \subset \op B^{*} \subset \op A^{*} + ``` + since $\op A \subset \op B$ implies $\op B^{*} \subset \opA^{*}$ by definition of the adjoint. + +- To summarize + - All symmetric extensions of $\opA$ are between $\opA$ and $\opA^*$. + - All closed and symmetric extensions of $\opA$ are between $\overline \opA$ and $\opA$. +""" + +# ╔═║ 77cde55f-7b10-4587-aebd-289eeff3d06d +md""" +The spectrum of a symmetric operator is already rather restricted. +Only 4 cases can arise : + +!!! note "Theorem 8 (Spectrum of symmetric operators)" + Let $\opA$ with $D(\opA)$ be a symmetric, closed operator. + Its spectrum is one of + + - the entire complex plane $\sigma(\opA)=\mathbb{C}$ + - the upper half plane $\sigma(\opA)=\mathbb{C}_{+}= \{ z \in \mathbb{C} \mid \im (z) \geq 0\}$ + - the lower half plane $\sigma(\op{A})=\mathbb{C}_{-}=\{z \in \mathbb{C} \mid \im (z) \leq 0\}$ + - non-empty and included in $\mathbb{R}$ : $\varnothing \neq \sigma(\opA) \subset \mathbb{R}$ + + In any case the spectrum never has eigenvalues in $\mathbb{C} \setminus \mathbb{R}$, meaning $\ker (\opA-z)=\{0\}$ if $\im (z) \neq 0$. + + Furthermore, if $\opA$ with $D(\opA)$ is symmetric and $z \in \mathbb{C} \setminus \mathbb{R}$ is such that $\opA - z$ is surjective then $\opA$ is closed and $z \in \resolvent(\opA)$, meaning that $\sigma(\opA)$ is restricted to the half-plane not containing $z$. + +""" + +# ╔═║ 0d1ee873-cb69-4e9b-ad7c-ccdf36ce4dad +md""" +- Just having the information that $\opA - z$ is surjective for a single $z$ with $\im (z) \neq 0$ is sufficient to exclude a half plane as the possible spectrum and implies $\opA = \bar \opA$. + + +- Further if $\opA$ symmetric and bounded over all $D(\opA)=\hilbert$ (e.g. finite dimensions), its spectrum is necessarily included in a ball of radius $\| \opA \|$. + To see that consider a $z \in \mathbb{C}$ with $|z|>\| \opA \|$ : + ```math + \left \| 1+ \frac{1}{z} (\opA - z ) \right \| = \frac{\| \opA \|}{|z|} <1 + ``` + So $-\big(\frac1z (\opA - z )\big)^{-1}= \left[1- \big(1+ \frac1z (\opA - z )\big) \right]^{-1}$ can be + computed by a convergent Neumann series making $-\frac1z (\opA-z)$ and thus $(\opA-z)$ invertible with bounded inverse, and thus $z \in \resolvent(\opA)$. + Therefore, $\sigma(\opA)$ is contained in a ball of radius $\| \opA \|$ leaving only the fourth case, i.e. $\sigma(\opA) \subset(-\| \opA \|,\| \opA \|) \subset \mathbb{R}$. +""" + +# ╔═║ 6dd9f4ed-6c02-4966-8397-5faad13f17d1 +md""" +- The proof of Theorem 8 is based on a simple equality : for all $x \in D(\opA)$ and $a, b \in \mathbb{R}$ we have + ```math + \begin{align} + \|( \opA -a-i b) x \|^{2} & = \langle (\opA -a-i b) x, (\opA -a-i b) x\rangle + \\ + & =\| (\opA -a) x \|^{2}+b^{2}\|x\|^{2}-2 b \im ( \langle (\opA-a) x, x \rangle ) + \\ + & =\| (\opA-a) x \|^{2}+b^{2} \| x \|^{2} + \end{align} + ``` + where we used that for symmetric operators + ```math + \langle( \opA-a) x, x\rangle=\langle x,(\opA-a) x\rangle=\overline{\langle(\opA-a) x, x \rangle} + ``` + is real. + + The resulting relation + ```math + \|( \opA -a-i b) x\|^{2} + =\|(\opA -a) x \|^{2}+b^{2}\|x\|^{2} + \geq b^{2}\|x\|^{2} + ``` + yields that, for $b \neq 0$, $(\opA -a-i b) x=0$ implies $x=0$, i.e. that there *cannot be non-real eigenvalues* for a symmetric operator. + + Furthermore, if the inverse $(\opA-a-i b)^{-1}$ exits it is automaticity bounded by + ```math + \|(A-a-i b)^{-1} \| \leq \frac{1}{|b|} + \tag{4} + ``` + + Therefore, for a symmetric operator, the only reason for $a+i b \in \sigma(\opA)$ is that $\opA -a-i b$ is not surjective. + + For the full proof of Theorem 8, see Lewin ([*ThΓ©rorie spectrale et mΓ©canique quantique*](https://doi.org/10.1007/978-3-030-93436-1) 2022), Theorem 2.23. +""" + +# ╔═║ 8f7acddd-9192-4b8b-a269-756194e46087 +md""" +### Self-Adjointess and Graph + +- We recall that an operator $\opA$ with $D(\opA) \subset \hilbert$ is *self-adjoint* if $\opA=\opA^{*}$, i.e. if $\opA$ is symmetric (implying $\opA \subset \op{A}^{*}$) and if $D (\opA^{*} )=D(\opA)$ + + +- Recall that for symmetric operators we have the graph inclusion + ```math + \graph(\opA) + = + \{(x, \opA x) \in D(\opA) \times \hilbert \} + \subset + \{(\opA y,-y) \in \hilbert \times D (\opA) \}^{\perp} + = + G (\opA^{*} ). + ``` + With self-adjointness, the inclusion becomes an equality : $G(\opA)=G (\opA^{*})$. + + +- As a result a symmetric operator is self-adjoint if and only if + ```math + \begin{align} + \langle x, \opA z\rangle + &= + \langle y, z \rangle && && \forall z \in D(\opA) + \\ + \Rightarrow x &\in D(\opA) && \text{ and } && \opA x=y. + \end{align} + ``` + +""" + +# ╔═║ ca779a00-60d2-434f-bca6-a387d9566783 +md""" +- Note that a self-adjoint operator is never the extension or restriction of a self-adjoint operator. + This goes back to our earlier realisation that $\op A \subset \op B$ implies $\op B^{*} \subset \op A^{*}$ thus $\op A \subset \op B \subset \op B^{*} \subset \op A^{*}$. + Therefore, $\opA=\opA^{*}$ implies $\opA=\op B$. + Crucially, one can therefore not *grow or shrink $D(\opA)$ preserving self-adjointness*. +""" + +# ╔═║ a31a2162-c299-4f2f-890e-cc450ebccf8f + + +# ╔═║ c1408ac8-a9c0-4834-bd7e-bb23dab01488 +TableOfContents() + +# ╔═║ 848e2e37-3605-4feb-9424-cdb76da54957 +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 395) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" + +[compat] 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"1fa38f19-a742-5d3f-a2b9-30dd87b9d5f8" +version = "1.3.7" + +[[deps.Hyperscript]] +deps = ["Test"] +git-tree-sha1 = "179267cfa5e712760cd43dcae385d7ea90cc25a4" +uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91" +version = "0.0.5" + +[[deps.HypertextLiteral]] +deps = ["Tricks"] +git-tree-sha1 = "7134810b1afce04bbc1045ca1985fbe81ce17653" +uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +version = "0.9.5" + +[[deps.IOCapture]] +deps = ["Logging", "Random"] +git-tree-sha1 = "b6d6bfdd7ce25b0f9b2f6b3dd56b2673a66c8770" +uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89" +version = "0.2.5" + +[[deps.InteractiveUtils]] +deps = ["Markdown"] +uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240" + +[[deps.JSON]] +deps = ["Dates", "Mmap", "Parsers", "Unicode"] +git-tree-sha1 = "31e996f0a15c7b280ba9f76636b3ff9e2ae58c9a" +uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6" +version = "0.21.4" + +[[deps.JuliaInterpreter]] +deps = ["CodeTracking", "InteractiveUtils", "Random", "UUIDs"] +git-tree-sha1 = "5d3a5a206297af3868151bb4a2cf27ebce46f16d" +uuid = "aa1ae85d-cabe-5617-a682-6adf51b2e16a" +version = "0.9.33" + +[[deps.LaTeXStrings]] +git-tree-sha1 = "50901ebc375ed41dbf8058da26f9de442febbbec" +uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +version = "1.3.1" + +[[deps.Latexify]] +deps = ["Format", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "OrderedCollections", "Requires"] +git-tree-sha1 = "5b0d630f3020b82c0775a51d05895852f8506f50" +uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" +version = "0.16.4" + + [deps.Latexify.extensions] + DataFramesExt = "DataFrames" + SymEngineExt = "SymEngine" + + [deps.Latexify.weakdeps] + DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" + SymEngine = "123dc426-2d89-5057-bbad-38513e3affd8" + +[[deps.LibCURL]] +deps = ["LibCURL_jll", "MozillaCACerts_jll"] +uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21" +version = "0.6.4" + +[[deps.LibCURL_jll]] +deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"] +uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0" +version = "8.4.0+0" + +[[deps.LibGit2]] +deps = ["Base64", "LibGit2_jll", "NetworkOptions", "Printf", "SHA"] +uuid = "76f85450-5226-5b5a-8eaa-529ad045b433" + +[[deps.LibGit2_jll]] +deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll"] +uuid = "e37daf67-58a4-590a-8e99-b0245dd2ffc5" +version = "1.6.4+0" + +[[deps.LibSSH2_jll]] +deps = ["Artifacts", "Libdl", "MbedTLS_jll"] +uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8" +version = "1.11.0+1" + +[[deps.Libdl]] +uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb" + +[[deps.LinearAlgebra]] +deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"] +uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" + +[[deps.Logging]] +uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" + +[[deps.LoweredCodeUtils]] +deps = ["JuliaInterpreter"] +git-tree-sha1 = "0b898aba6cb0b01fb96245fa5375accb651a241a" +uuid = "6f1432cf-f94c-5a45-995e-cdbf5db27b0b" +version = "3.0.0" + +[[deps.MIMEs]] +git-tree-sha1 = "65f28ad4b594aebe22157d6fac869786a255b7eb" +uuid = "6c6e2e6c-3030-632d-7369-2d6c69616d65" +version = "0.1.4" + +[[deps.MacroTools]] +deps = ["Markdown", "Random"] +git-tree-sha1 = "2fa9ee3e63fd3a4f7a9a4f4744a52f4856de82df" +uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09" +version = "0.5.13" + +[[deps.Markdown]] +deps = ["Base64"] +uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" + +[[deps.MbedTLS_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1" +version = "2.28.2+1" + +[[deps.Mmap]] +uuid = "a63ad114-7e13-5084-954f-fe012c677804" + +[[deps.MozillaCACerts_jll]] +uuid = "14a3606d-f60d-562e-9121-12d972cd8159" +version = "2023.1.10" + +[[deps.NetworkOptions]] +uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908" +version = "1.2.0" + +[[deps.OpenBLAS_jll]] +deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"] +uuid = "4536629a-c528-5b80-bd46-f80d51c5b363" +version = "0.3.23+4" + +[[deps.OrderedCollections]] +git-tree-sha1 = "dfdf5519f235516220579f949664f1bf44e741c5" +uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d" +version = "1.6.3" + +[[deps.Parsers]] +deps = ["Dates", "PrecompileTools", "UUIDs"] +git-tree-sha1 = "8489905bcdbcfac64d1daa51ca07c0d8f0283821" +uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" +version = "2.8.1" + +[[deps.Pkg]] +deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] +uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" +version = "1.10.0" + +[[deps.PlutoHooks]] +deps = ["InteractiveUtils", "Markdown", "UUIDs"] +git-tree-sha1 = "072cdf20c9b0507fdd977d7d246d90030609674b" +uuid = "0ff47ea0-7a50-410d-8455-4348d5de0774" +version = "0.0.5" + +[[deps.PlutoLinks]] +deps = ["FileWatching", "InteractiveUtils", "Markdown", "PlutoHooks", "Revise", "UUIDs"] +git-tree-sha1 = "8f5fa7056e6dcfb23ac5211de38e6c03f6367794" +uuid = "0ff47ea0-7a50-410d-8455-4348d5de0420" +version = "0.1.6" + +[[deps.PlutoTeachingTools]] +deps = ["Downloads", "HypertextLiteral", "LaTeXStrings", "Latexify", "Markdown", "PlutoLinks", "PlutoUI", "Random"] +git-tree-sha1 = "5d9ab1a4faf25a62bb9d07ef0003396ac258ef1c" +uuid = "661c6b06-c737-4d37-b85c-46df65de6f69" +version = "0.2.15" + +[[deps.PlutoUI]] +deps = ["AbstractPlutoDingetjes", "Base64", "ColorTypes", "Dates", "FixedPointNumbers", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "MIMEs", "Markdown", "Random", "Reexport", "URIs", "UUIDs"] +git-tree-sha1 = "ab55ee1510ad2af0ff674dbcced5e94921f867a9" +uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +version = "0.7.59" + +[[deps.PrecompileTools]] +deps = ["Preferences"] +git-tree-sha1 = "5aa36f7049a63a1528fe8f7c3f2113413ffd4e1f" +uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a" +version = "1.2.1" + +[[deps.Preferences]] +deps = ["TOML"] +git-tree-sha1 = "9306f6085165d270f7e3db02af26a400d580f5c6" +uuid = "21216c6a-2e73-6563-6e65-726566657250" +version = "1.4.3" + +[[deps.Printf]] +deps = ["Unicode"] +uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7" + +[[deps.REPL]] +deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"] +uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb" + +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.Reexport]] +git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" +uuid = "189a3867-3050-52da-a836-e630ba90ab69" +version = "1.2.2" + +[[deps.Requires]] +deps = ["UUIDs"] +git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7" +uuid = "ae029012-a4dd-5104-9daa-d747884805df" +version = "1.3.0" + +[[deps.Revise]] +deps = ["CodeTracking", "Distributed", "FileWatching", "JuliaInterpreter", "LibGit2", "LoweredCodeUtils", "OrderedCollections", "Pkg", "REPL", "Requires", "UUIDs", "Unicode"] +git-tree-sha1 = "677b65e17aeb6b4a0be1982e281ec03b0f55155c" +uuid = "295af30f-e4ad-537b-8983-00126c2a3abe" +version = "3.5.16" + 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\ No newline at end of file diff --git a/10_Spectra_self_adjoint.jl b/10_Spectra_self_adjoint.jl new file mode 100644 index 0000000..c8ac27d --- /dev/null +++ b/10_Spectra_self_adjoint.jl @@ -0,0 +1,2173 @@ +### A Pluto.jl notebook ### +# v0.20.3 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + #! format: off + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end + #! format: on +end + +# ╔═║ 4004f2e8-5738-47e2-9d3c-8b68b896b9a5 +begin + import TikzPictures.TikzPicture + using LaTeXStrings + using PlutoUI + using HypertextLiteral + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ 565cdda1-d844-4bed-8f48-fe57c1428136 +begin + using QuadGK + using LinearAlgebra + + # basis functions and their Laplacian + basis_function(x,n) = sin(x * n) * sqrt(2/Ο€) + laplacian(x,n) = - n^2 * sin(x * n) * sqrt(2/Ο€) + + # potential + function gaussian(x) + A = -1000 + sigma = Ο€/16 + return A * exp(-((x - Ο€/2)/sigma)^2) + end + + #matrix elements + function Mij(i,j,V) + # I know that this is definitely not the most efficient approach from a computational standpoint, but it works + quad = quadgk(x -> conj(basis_function(x,i)) * (- laplacian(x,j) / 2 + V(x) * basis_function(x,j)), 0, Ο€, atol=1e-6) + return quad[1] + end + + # reference values + eig = let + n = 256 + M = zeros(ComplexF64,n,n) + for i in 1:n + for j in i:n + M[i,j] = Mij(i,j,gaussian) + end + end + M = Hermitian(M) + eig = eigen(M) + end + + # varying basis size + n_range = 2:2:80 + eigenvalues = ones(Union{Missing,Float64},maximum(n_range),length(n_range)) + eigenvalues .= missing + for (i,n) in enumerate(n_range) + M = zeros(ComplexF64,n,n) + for i in 1:n + for j in i:n + M[i,j] = Mij(i,j,gaussian) + end + end + M = Hermitian(M) + eigenvalues[1:n,i] = eigen(M).values + end +end + +# ╔═║ 1478c459-0933-4c3f-8e84-86d051bea4b3 +md"# Spectra of Self-Adjoint Operators" + + +# ╔═║ 21814beb-9d27-44c7-a41f-1191d856d1b7 +md""" + +## Self-adjoint operators + +The following strong result is the justification for the elaborate construction of self-adjoint operators, where great care in constructing their domain is often needed. + +!!! note "Theorem 1" + Let $\opA$ be a symmetric operator with domain $D(\opA) \subset \hilbert$, + that is + ```math + \begin{align} + \langle \varphi, \opA \varphi \rangle & = \langle \opA \varphi, \varphi \rangle && \forall \varphi \in D(\opA) + \end{align} + ``` + Then the following statements are equivalent + + 1. $\opA$ is self-adjoint, i.e. $D(\opA^*) = D(\opA)$ + + 2. $\varnothing \neq \sigma(\opA) \subset \mathbb R$ + + 3. $\exists \lambda \in \mathbb C$ such that $\opA - \lambda$ and + $\opA - \bar \lambda$ are both surjective from $D(\opA) \to \hilbert$. + +We thus obtain that the familiar result that self-adjoint operators have a real spectrum. + +""" + +# ╔═║ d2b95a1b-5e4c-4e85-8a6e-d893c9c3cbde +md""" +## Weyl sequences for spectral characterization + +Another important characterisation method is based on the convergence of bounded sequences, where we already noted differences between the finite and infinite dimensional case. +To fully appreciate the details we need a few more definitions. +""" + +# ╔═║ 206bf4ec-9bdd-4141-8ee4-570f077c06d0 +md""" + + + +!!! note "Definition 1 (Weak Convergence)" + Let $\hilbert$ be a Hilbert + space. A sequence $(\phi_n) \in \hilbert$ is said to *converge weakly* + against a $\varphi \in \hilbert$ if + ```math + \begin{align} + \lim_{n \to \infty} \langle \phi_n , f \rangle &= \langle \varphi , f \rangle && \forall f \in \mathcal \hilbert. + \end{align} + ``` + In this case we also write + $\phi_n \rightharpoonup \varphi$. + + +!!! tip "Remark (Strong convergence implies weak convergence)" + Let + $\phi_n \to \varphi$ strongly in $\hilbert$. + Then, it holds + $\| \phi_n - \varphi \| \to 0$ as $n \to \infty$. + Let further $f \in \hilbert$. We obtain + ```math + \lim_{n \to \infty} | \langle \phi_n - \varphi, f \rangle | \leq \lim_{n \to \infty} \| \phi_n - \varphi \| \cdot \| f \| = 0 + ``` + Hence, strong convergence implies weak convergence. + + +!!! tip "Remark" + If $\phi_n \rightharpoonup \varphi$ weakly and + $\| \phi_n \| \to \| \varphi \|$ strongly, then $\phi_n \to \varphi$ strongly + because + ```math + \begin{align} + \| \phi_n - \varphi \| ^2 &= \| \phi_n \| ^2 - 2 \langle \varphi, \phi_n \rangle + \| \varphi \|^2 + + \end{align} + ``` + Which gives, as $n \to \infty$ + ```math + \begin{align} + 2 \| \varphi \| ^2 - 2 \langle \varphi, \varphi \rangle & = 0 + + \end{align} + ``` + + + + +""" + +# ╔═║ 390974fb-36d3-44e1-8543-24a16e400512 +md""" +In infinite dimensions closed and bounded sets are no longer necessarily compact. +Thus bounded sequences may not have strongly converging subsequences. +However, + +!!! note "Theorem 2" + Let $\hilbert$ be a Hilbert space and + $(\phi_n) \subset \hilbert$ a bounded sequence. Then, there always exists a + *weakly* convergent subsequence. + + +!!! warning "Example 1" + Consider the sequence $(e_i) = (0,0,\dots,0,1,0,0,\dots)$, the + sequence of unit vectors in $\ell^2 (\mathbb R)$. + The + sequence is bounded, as each element is inside the infinite dimensional unit ball ($\| e_i \| = 1$), and no strongly convergent subsequence exists. + However, + $e_i \rightharpoonup 0$. + +With this in mind we return to the characterization of the spectra of self-adjoint operators. + +!!! note "Definition 2 (Weyl Sequence)" + A sequence $(\phi_n) \subset D(\opA)$ + with $\| \phi_n \| = 1$, which satisfies + $\| (\opA - \lambda) \phi_n \| \to 0$ for a $\lambda \in \mathbb R$ is called a + *Weyl sequence*. + + + +""" + +# ╔═║ 95acb6bd-42d7-4ede-b425-a91ed9640d9a +md""" +!!! note "Theorem 3" + Let $\opA$ be self-adjoint with $D(\opA) \subset \hilbert$ and + $\lambda \in \mathbb R$. The following are equivalent + + 1. $\lambda \in \sigma(\opA)$ + + 2. $\inf_{\varphi \in D(\opA), \| \varphi \| = 1} \| (\opA - \lambda) \varphi \| = 0$ + + 3. There exists a Weyl sequence for $\lambda$. + + +!!! tip "Remark" + The third statement explains nicely a key difference between finite and infinite dimensions. + + - **In finite dimensions**, the unit sphere is compact, since it is closed and bounded. + Therefore, for each Weyl sequence $\phi_n$ with $\| \phi_n \| = 1$, we can extract a convergent subsequence + $\phi_{n_k} \to \varphi \in \hilbert$. + + Since all operators are bounded / continuous in finite dimensions, $\opA \phi_{n_k} \to \opA \varphi$. + Thus, $\| (\opA-\lambda) \phi_{n_k} \| \to 0$ implies $\opA \varphi = \lambda \varphi$. + + Furthermore, from the triangle inequality + ```math + \begin{align} + \bigl \vert \| \phi_{n_k} \| - \| \phi_{n_k} - \varphi \| \bigr | \leq \| \varphi \| \leq \| \phi_{n_k} \| + \| \phi_{n_k} - \varphi \| + + \end{align} + ``` + In addition, since $\phi_{n_k} \to \varphi$ strongly and $\| \phi_n \| = 1$ these three statements imply $\| \varphi \| = 1$. + + A Weyl sequence for $\lambda$ thus always yields an eigenpair and $\sigma(\opA)$ consists only of eigenvalues. + Moreover *Weyl sequences converge (strongly) to eigenvalues.* + + + - **In infinite dimensions**, $\| \phi_n \| = 1$ implies that we have a bounded sequence. + Because of Theorem 2, there exists a bounded subsequence $\phi_{n_k}$ with $\phi_{n_k} \rightharpoonup \varphi$ weakly. + Further, the strong convergence $\| (\opA - \lambda) \phi_n \| \to 0$ implies weak convergence. Thus, for any $f \in D(\opA)$, it holds + + ```math + \begin{align} + 0 &= \lim_{n\to \infty} \langle f, (\opA - \lambda) \phi_n \rangle \\ + &= \lim_{n \to \infty} \langle (\opA - \lambda) f , \phi_n \rangle \\ + &= \langle (\opA - \lambda) f, \varphi \rangle + + \end{align} + ``` + where we used the symmetry of $\opA - \lambda$ and the weak convergence of $\phi_n$. + Therefore, + ```math + \begin{align} + \tag{1} + \langle \opA f , \varphi \rangle &= \lambda \langle f, \varphi \rangle && \forall f \in D(\opA) + \end{align} + ``` + + Next, we argue based on the graph of $\opA$ + ```math + \begin{align} + G(\opA) &= \{ ( \varphi, \opA \varphi ) \in \hilbert \times \hilbert \mid \varphi \in D(\opA) \} + \end{align} + ``` + and the graph of the adjoint, which can be written + + ```math + \begin{align} + G(\opA^*) &= \{ ( g, \opA^* g ) \in \hilbert \times \hilbert \mid g \in D(\opA^*) = D(\opA) \} \\ + &= \{ (g,h) \in \hilbert \times \hilbert \mid \langle \opA f, g \rangle = \langle f, h \rangle \ \forall f \in D(\opA) \}. + \end{align} + ``` + Employing (1), we deduce by comparing the expressions that $( \varphi, \lambda \varphi ) \in G(\opA^*)$. + The only element for which this is possible is $(g , \opA ^* g ) = ( \varphi, \lambda \varphi )$. + + Hence, + ```math + \begin{align} + \lambda \varphi = \opA^* \varphi = \opA \varphi + \end{align} + ``` + using the self-adjointness of $\opA$. + + Consider the case where $\lambda \in \sigma(\opA)$, but $\lambda$ is *not* an eigenvalue. + Then $\mathop{\mathrm{Ker}}(\opA - \lambda) = \{ 0 \}$ and it must hold $\phi_n \rightharpoonup \varphi = 0$. + *Weyl sequences converge weakly to zero.* + +""" + +# ╔═║ 20958c09-7372-4c0b-bd3a-3011758cba2e +md""" +One consequence of Theorem 3 is : + +!!! note "Theorem 4" + Let $\opA$ be a self-adjoint operator on + $D(\opA) \subset \hilbert$. Then + ```math + \begin{align} + \inf \sigma (\opA) &= \inf_{0 \neq \varphi \in D(\opA)} \frac{\langle \varphi, \opA \varphi \rangle}{\langle \varphi, \varphi \rangle} + \\ + \sup \sigma(\opA) &= \sup_{0 \neq \varphi \in D(\opA)} \frac{\langle \varphi, \opA \varphi \rangle}{\langle \varphi, \varphi \rangle} + + \end{align} + ``` + +In particular, $\sigma(\opA) \subset [a, \infty )$ with +$a > - \infty$ if and only if +$\langle \varphi, \opA \varphi \rangle \geq a \| \varphi \| ^2$ for all $\varphi \in D(\opA)$. +Such operators are called *bounded from below*. +Similarly, upper semi-bounded operators satisfy $\langle \varphi, \op B \varphi \rangle \leq b$ and satisfy $\sigma (\op B) \subset (- \infty , b]$ + +!!! tip "Remark" + The striking similarity with Courant-Fisher for matrices already points to a possible generalization of the min-max statement statements for operators. + This we will pursue next. + + + +""" + +# ╔═║ 2f377ee6-e23e-432f-aeb6-9097fa260027 +md""" +Beforehand, let us pause for a second and use this result to deduce the spectra for a few self-adjoint operators on $\hilbert = L^2(\mathbb R ^d)$ + +!!! warning "Example 2 (Identity operator)" + $\mathop{\mathrm{id}}: \hilbert \to \hilbert$ (i.e. $D(\mathop{\mathrm{id}}) = \hilbert$) is clearly bounded from above and below by 1. + Thus, $\sigma(\mathop{\mathrm{id}}) = \{ 1\}$ + + +!!! warning "Example 3 (Multiplication by a continuous function)" + Let + $V : \mathbb R^d \to \mathbb R, \op V : D(\op V) \to \hilbert$ where $D(\op V) = \{ f \in L^2 (\mathbb R^2) \mid Vf \in L^2 (\mathbb R^2) \}$. + Again, $\op V$ is bounded from below by $\inf_{x \in \mathbb R^d} V(x)$ and from above by + $\sup_{x \in \mathbb R^d} V(x)$. Therefore, + ```math + \begin{align} + \sigma(V) = \overline{\mathop{\mathrm{range}}(V)} = \left [ \inf_{x\in \mathbb R^3} V, \sup_{x \in \mathbb R^3} V \right ] + + \end{align} + ``` + + +!!! warning "Example 4 (Laplace operator - Ξ”)" + We want to show explicitly that $\sigma ( - \laplacian) = [ 0 , \infty )$ for the Laplace operator $- \laplacian$ with + $D(- \laplacian) = H^2 (\mathbb R^d)$. + We already stated this operator to be self-adjoint. + Using integration by parts, we find + ```math + \begin{align} + \forall \varphi \in H^2(\mathbb R^d) && \langle \varphi, - \laplacian \varphi \rangle = \langle \nabla \varphi, \nabla \varphi \rangle \geq 0. + \end{align} + ``` + Thus $\sigma(- \laplacian) \subset [0, \infty)$. + + Now, we take $k_0 \in \mathbb R^d, f \in H^2(\mathbb R^d)$ and define a Weyl sequence + ```math + f_n(x) = n^{-d/2} f(x/n) e^{i x \cdot k_0} + ``` + with Fourier transform + ```math + \hat f_n(k) = n^{d/2} \hat f (n (k - k_0)). + ``` + Using $| \cdot |$ to denote norms in $\mathbb R^d$ : + ```math + \begin{align} + \| (- \laplacian - |k_0| )^2 f_n \| ^2 &= \int_{\mathbb R^d} \bigl | |k|^2 - |k_0|^2 \bigr | ^2 |\hat f_n (k) |^2 dk + \\ + &= \int_{\mathbb R^d} \left | \left | k_0 + \frac{p}{n} \right |^2 - |k_0|^2 \right |^2 |\hat f (p) |^2 dp + \\ + &=\frac1{n} \int_{\mathbb R^d} \left | 2 p \cdot k_0 + \frac{|p|^2}{n} \right |^2 |\hat f (p) |^2 dp + \end{align} + ``` + which converges to 0 as $n \to \infty$. + Therefore (Theorem 3) $|k_0|^2 \in \sigma(- \laplacian).$ + As $k_0$ spans $\mathbb R^d$, $|k_0|^2$ spans $[0, \infty)$, so $[0, \infty) \subset \sigma(- \laplacian)$. + + Therefore, we have $[0,\infty) \subset \sigma(-\laplacian) \subset [0, \infty)$. + Hence, $[0, \infty) = \sigma(-\laplacian)$ +""" + +# ╔═║ d2bf63fb-6035-49b2-9182-c606bb3643af +md""" +We already introduced the concept of an eigenvalue as well as the point spectrum $\sigma_p$ : +```math +\begin{align} + \sigma_p(\opA) &= \{ \lambda \in \mathbb C \mid \lambda \text{ is an eigenvalue of } \opA \} + \\ + &= \{ \lambda \in \mathbb C \mid \mathrm{Ker}(\opA - \lambda) \neq \{0\} \} +\end{align} +``` +A natural characterization of the spectrum is to spit $\sigma (\opA) = \sigma_p (\opA) \ \dot \cup \ \sigma_{\text{cont}} (\opA)$ where + +!!! note "Definition 3 (Continuous spectrum)" + Let $\opA : D(\opA) \to \hilbert$ be a self-adjoint operator. + We call the *continuous spectrum* $\sigma _{\text{cont}}$ + ```math + \begin{align} + \sigma_{\text{cont}} (\opA) = \sigma \setminus \sigma_p (\opA) + + \end{align} + ``` + +This classification turns out to be not very useful for considering spectral approximations as small perturbation of $\opA$ can easily mix $\sigma_p$ and $\sigma_{\text{cont}}$. +Whe therefore use an alternate : + + + +""" + +# ╔═║ a803317a-07c5-4f95-bc45-4c24f3971385 +md""" +!!! note "Definition 4 (Isolated point)" + Let $\opA : D(\opA) \to \hilbert$ be a self-adjoint operator. + An entry + $\lambda \in \sigma(\opA)$ is called an *isolated point* if there exists $\varepsilon > 0$ such that + ```math + \begin{align} + [\lambda - \varepsilon, \lambda + \varepsilon] \cap \sigma(\opA) = \{ \lambda \} + \end{align} + ``` + +One can show that every isolated point of $\sigma(\opA)$ is an eigenvalue. +They are thus interesting entries, motivating the following spectrum : + + +!!! note "Definition 5 (Discrete and essential spectrum)" + Let $\opA : D(\opA) \to \hilbert$ be a self-adjoint operator. + ```math + \sigma_{\text{disc}} = \{ \lambda \in \sigma(\opA) \mid \lambda \text{ is an isolated point and has finite multiplicity} \} + ``` + is called the *discrete spectrum*. + + The complement + ```math + \sigma_{\text{ess}} = \sigma(\opA) \setminus \sigma_{\text{disc}} (\opA) + ``` + is called the *essential spectrum*. + + +""" + +# ╔═║ f2b8a3ba-71e9-48a9-b77d-91aa844f3da3 +md""" +Let us illustrate these concepts by sketching the spectrum of SchrΓΆdinger operators $\opH = - \laplacian / 2 + V$ on $\hilbert = L^2(\mathbb R^3)$. +We take $V \in L^2(\mathbb R^3) + L^\infty_\varepsilon(\mathbb R^3)$, that is to say that the potential needs to be split into a square integrable ($L^2$) short-range part and a bounded long-range part that also is negligible at infinity ($L^\infty_\varepsilon$). +For example, the Coulomb potential satisfies this. +Since $L^\infty_\varepsilon \subset L^\infty$, $\opH$ is self-adjoint using $D(\opH) = H^2(\mathbb R^3)$. + +Then one can show +```math + \sigma_{\text{ess}} (\opH) = [0, \infty). +``` +Therefore, the spectrum looks like this : + +--- +""" + + +# ╔═║ dda55c4f-1a71-477b-b460-0c67629c2904 +TikzPicture(L""" +% %real line + % \draw[>=latex,->] (-5,-5) -- (5,0) node[right]{$\mathbb R$}; + % \draw (0,0) node{$|$} ; + % \draw (0,0.1) node[above]{0}; + + %sigma_p + \foreach \time [evaluate=\time as \energy using {- 5 / \time^1 + 5/30} ] in {1,...,30}{ + \draw (\energy,-1.5) node{$\times$}; + } + + \foreach \energy in {2,2.5,3.5,0}{ + \draw (\energy,-1.5) node[color=blue]{$\times$}; + } + + \draw (5,-1.5) node[right]{$\sigma_p$}; + + \draw[->] (0,-1) node[above]{$\infty$ multiplicity} -- (0,-1.35); + + \foreach \energy in {2,2.5,3.5}{ + \draw[->,blue] (\energy,-1) -- (\energy,-1.35); + } + + \draw[blue] (3,-1) node[above,blue]{Non-isolated eigenvalues} ; + + %sigma_cont + + \draw (0,-2) node[blue]{$($} -- (5,-2) node[right]{$\sigma_{\textrm{cont}}$}; + + \foreach \energy in {2,2.5,3.5}{ + \draw (\energy,-2) node[color=white]{$\bullet$} node[blue]{$\circ$} ; + } + + %sigma_disc + \foreach \time [evaluate=\time as \energy using {- 5 / \time^1 + 5/30} ] in {1,...,30}{ + \draw (\energy,-3) node{$\times$}; + } + + \draw (5,-3) node[right]{$\sigma_{\textrm{disc}}$}; + + %sigma_ess + \draw (0,-3.5) node[blue]{$[$} -- (5,-3.5) node[right]{$\sigma_{\textrm{ess}}$}; +""",width="20cm",options="scale=1.1") + +# ╔═║ 8e5dd330-181a-4279-b54b-47be263e13a9 +md""" +--- + +Notice how eigenvalues (members of $\sigma_p$) may be embedded inside the continuous spectrum. +These are very unstable to perturbations (so-called resonance states) and, as we will see now, hard to approximate. + +Notice that, unlike $\sigma_{\text{cont}}$, $\sigma_{ess}$ is always closed for a self-adjoint operator. + + +""" + +# ╔═║ e60a51b3-88ed-4e03-9937-4c86773e33e8 +md""" +Overall, Weyl sequences turn out to yield a useful characterization of the spectra of self-adjoint operators, and overview of which is shown here : + +Spectrum | Weyl sequence for spectral characterization +---|:--- +$\lambda \in \sigma(\opA)$ | $\exists \text{ a Weyl sequence } (\phi_n) \subset D(\opA) \text{ s.t. } \| \phi_n \| = 1 \text{ and } \| (\opA - \lambda) \phi_n \| \to 0$ +$\lambda \in \sigma_{\text{ess}} (\opA)$ | $\exists \text{ a Weyl sequence s.t. } \phi_n \rightharpoonup 0 \text{ (weakly)}$ +$\lambda \in \sigma_{\text{disc}} (\opA)$ | $\text{\emph{All} Weyl sequences have subsequences } \phi_{n_k} \to \varphi \text{ (strongly)}$ +$\lambda \in \sigma_{\text{cont}} (\opA)$ | $\text{\emph{All} Weyl sequences verify } \phi_n \rightharpoonup 0 \text{ (weakly)}$ +$\lambda \in \sigma_{\text{p}}(\opA)$ | $\exists \text{ a Weyl sequence with a weak limit different from 0, i.e. } \mathop{\mathrm{Ker}}(\opA - \lambda) \neq 0.$ +""" + +# ╔═║ 1b178edd-98f9-4533-b3df-38efcd026333 +md""" +## Courant-Fisher + +To finish up our discussion we will now work towards a generalization of the Courant-Fisher characterization. +For this, we once again consider the quadratic form $q_\opA(u)$ +```math + q_\opA (u) = \langle u, \opA u \rangle +``` +and the sequilinear form +```math + a_\opA (u,v) = \langle u, \opA v \rangle +``` +which we had already encountered in the context of matrices. + +A priori, these are only defined for $u,v \in D(\opA)$. +However, in the following paragraph we introduce the *form domain* $Q(\opA)$, an extension of $D(\opA)$ + +If $\opA$ is bounded from below then there exists $\alpha > -\infty$ such that +```math + q_A (u) \geq \alpha \| u \| ^2 +``` +Note that is the case for all SchrΓΆdinger operators we consider. +For simplicity, we take $\alpha > 0$ (otherwise consider a shifted operator $\tilde \opA = \opA - \alpha + \varepsilon$ with $\varepsilon >0$). +Then +```math + \begin{align} + \langle u,v \rangle_\opA \equiv \langle u, \opA v \rangle && u,v \in D(\opA) + \end{align} +``` +is an inner product with induced norm +```math +\begin{align} + \| u \|_A \equiv q_A(u) > 0 && u \in D(\opA) \tag{Energy norm} +\end{align} +``` + $D(\opA)$ is not complete with respect to $\| \cdot \|_\opA$. +Therefore, the idea is now to consider sequences $(u_n)_n \subset D(\opA)$, which are clearly Cauchy with respect to $\| \cdot \|_\opA$, and extend the definition, i.e. if $u_n \to u$ then we demand $q_\opA (u_n) \to q_\opA (u)$. +In this way, we can extend the definition *uniquely* to $a_\opA (u,v)$, $q_\opA(u)$ with $u,v \in Q(\opA)$, called the *form domain* of $\opA$, and which satisfies +```math +\begin{align} + D(\opA) \leqslant Q(\opA) \leqslant \hilbert +\end{align} +``` +Where in each case the embedding is dense and continuous (Lewin [*ThΓ©rorie spectrale et mΓ©canique quantique*](https://doi.org/10.1007/978-3-030-93436-1) 2022, Theorem 3.10). + +""" + +# ╔═║ 819cccb8-fe5b-4bf6-9998-33ce4ab30819 +md""" + +The importance of the form domain is that is allows for a weak formulation : + +!!! note "Theorem 5" + Let $\opA$ be self-adjoint. The following are equivalent + + 1. $\varphi \in D(\opA)$ and $\opA \varphi = \lambda \varphi$. (*Strong formulation*) + + 2. $\varphi \in Q(\opA)$ and $\exists \lambda \in \mathbb R$ s.t. + $a_A(f,\varphi) = \lambda \langle f, \varphi \rangle$ for all $f \in Q(\opA)$. (*Weak + formulation*) + + + +""" + +# ╔═║ 68f786f1-45a4-475b-a0bb-37636819ddf2 +md""" +Based on this formulation we obtain, again based on Weyl sequences : + +!!! note "Theorem 6" + Let $\opA$ be a self-adjoint operator with form domain $Q(\opA) \subset \hilbert$, then + the *bottom of the essential + spectrum* $\Sigma(\opA)$ is uniquely defined by + ```math + \begin{align} + \Sigma(\opA) \coloneqq \min \sigma_{\text{ess}} (\opA) = \min_{\substack{(v_n) \in Q(\opA)^\mathbb N \\ \| v_n \| = 1 \\ v_n \rightharpoonup 0}} \liminf_{n \to \infty} q_A(v_n) + + \end{align} + ``` + with the convention $\Sigma(\opA) = \infty$ if and only if $\sigma_{\text{ess}}(\opA) = \varnothing$. + +!!! tip "Remark" + Typical form domains use again Sobolev spaces. For example + ```math + \begin{align} + Q(- \laplacian) &= H^1(\mathbb R^d) && \text{on } L^2(\mathbb R^d) + \\ + Q(- \laplacian + V) &= H^1(\mathbb R^3) && \text{on } L^2(\mathbb R^3) \text{ with } V \text{ appropriate} + \end{align} + ``` + +""" + +# ╔═║ f9f9e20a-0beb-46c6-863b-789d04d0b2af +md""" +Finally, we can state the equivalent the min-max principle that we already saw for matrices, extended to the case of operators. + +!!! note "Theorem 7 (Courant-Fisher)" + Let $\opA$ be self-adjoint and bounded + from below with form domain $Q(\opA) \leqslant \hilbert$. + Then + ```math + \begin{align} + \mu_k (\opA) \coloneqq \inf_{\substack{W \subset Q(\opA) \\ \dim (W) = k}} \max_{\substack{\varphi \in W \\ \| \varphi \|_\hilbert = 1}} q_A(\varphi) = \inf_{\substack{W \subset Q(\opA) \\ \dim (W) = k}} \max_{0 \neq \varphi \in W} R_A(\varphi) + + \end{align} + ``` + is equal to + + 1. The k$^{\textsf{th}}$ eigenvalue (counting multiplicities) of $\opA$ if $\opA$ has at + least $k$ eigenvalues below $\Sigma(\opA).$ + + 2. Otherwise, $\mu_k(\opA) = \Sigma(\opA)$. + + We re-introduced the *Rayleigh quotient*, this time for operators + ```math + R_\opA (\varphi ) \coloneqq \frac{q_\opA (\varphi)}{\| \varphi \|^2_\hilbert } + ``` + +One way to check for the existence of $k$ eigenvalues below $\Sigma(\opA)$ is : + +!!! tip "" + If there is a subspace $W \leq Q(\opA)$ with $\text{dim} W = k$, such that + ```math + \max_{0 \neq \varphi \in W} R_\opA (\varphi) < \Sigma(\opA) + ``` + then there are at least k eigenvalues. + +In other words, as long as $\mu_k(\opA) < \Sigma(\opA)$, the situation is as for Hermitian matrices. +As soon as $k$ eigenvalues have been found, we can no longer learn anything about $\sigma(\opA)$ by increasing the subspace size. + +This leads to a severe restriction for the numerical approximation of eigenspectra, as we are in most cases restricted to below $\Sigma(\opA)$. +However, in this case we are able to estimate eigenspectra, and obtain error bounds in a manner similar to that of matrices. + +""" + +# ╔═║ 9ccec1c6-55e8-4db6-aa70-18dabc87eca8 +md""" +## Approximation of $\lambda_k < \Sigma(\opA)$ + +A straightforward technique to approximate the eigenspace of operators is to employ a simple *projection technique*, i.e. to employ a *finite-dimensional subspace* $S \subset Q(\opA)$ in the infimum. +The idea is the same as we discussed in the context of diagonalization algorithms. +*Assuming* $\mu_k(\opA) < \Sigma(\opA)$, i.e. that $\opA$ has $k$ eigenvalues $\lambda_1, \dots, \lambda_k$ (counting multiplicities) below $\Sigma(\opA)$ yields +```math +\begin{align} +\tag{2} + \lambda_k = \mu_k (\opA) = \inf_{\substack{W \subset {\color{blue}Q(\opA)} \\ \dim (W) = k}} \max_{0 \neq \varphi \in W} R_A(\varphi) \leq \inf_{\substack{W \subset {\color{blue} S} \\ \dim (W) = k}} \max_{0 \neq \varphi \in W} R_A(\varphi) +\end{align} +``` + + +- Assuming $S$ to be $d$-dimensional, and taking a basis $\mathbb B = (\chi_1, \dots, \chi_d)$, we realize that the right-hand side of (2) to be nothing else than the $k$-th eigenvalue of the matrix + ```math + \begin{align} + (M_S^\opA)_{ij} = q_A(\chi_i, \chi_j) \cong \langle \chi_i , \opA \chi_j \rangle + \end{align} + ``` + Note that the second equality is only fully correct for $S \subset D(\opA)$. In other words + ```math + \begin{align} + \lambda_k (\opA) = \mu_k(\opA) \leq \lambda_k (M_S^\opA) + \end{align} + ``` +""" + +# ╔═║ f743838b-4204-432e-a81e-77af58737cba +md""" +- In practice one then keeps augmenting the subspace $S$, e.g. by increasing the approximation basis $\mathbb B$ following the principle + ```math + \lambda_k(\opA) = \mu_k(\opA) = \inf_{\substack{S \subset Q(\opA) \\ \dim Q(\opA) \geq k}} \lambda_k (M_S^\opA) + ``` + +- Notice, however, how this technique is *unable to approximate above $\Sigma(\opA)$.* + In other words, *some eigenvalues* of $\opA$, those embedded in $\sigma_{\text{ess}} (\opA)$ *cannot be found* using this projection technique. +""" + +# ╔═║ e80b1728-592a-4a97-b4ab-20b3ef8604c4 +md""" +--- +#### Illustration + +Eigenenergies of the 1D gaussian potential well, i.e. the spectrum of the following SchrΓΆdinger operator +```math +\mathcal H = - \frac1{2} \frac{d^2}{dx^2} - A e^{-(x/\sigma)^2} +``` +""" + +# ╔═║ 843f48c0-b38f-4c94-8539-cb0ca1a85252 +md""" +y axis limits (slide to zoom in/out) $\qquad$ $(@bind ylim PlutoUI.Slider([2000,1000,500,200,100,50,25]; default=1000, show_value=true)) +""" + +# ╔═║ f41ef893-8910-4e6b-9f08-cccba464bac1 +#plotting +begin + using Plots + + plot(ylims=(-ylim,ylim),ylabel="Calculated eigenvalues",xlabel="Dimension of discrete basis") + + hline!([0],color=:black,label=nothing) + for i in 1:maximum(n_range) + ydata = eigenvalues[i,:] + color = :gray + if minimum(skipmissing(ydata)) < 0 + color = i + hline!([eig.values[i]],color=i,linestyle=:dash,label=nothing,linewidth=0.5) + end + plot!(n_range,ydata,marker=:x,label=nothing,color=color,linewidth=0.5) + end + plot!() +end + +# ╔═║ 59832711-e53c-405c-b83d-3dc4fc05fa53 +md""" + +--- +## Kato-Temple bound + +To close our discussion on the spectra of operators, we want to rationalize why the Kato-Temple bound can still be employed for the eigenpairs below the essential spectrum. +We restate : + +""" + +# ╔═║ f950d40e-6ec7-4ee5-81c3-95314b52d029 +md""" +!!! note "Theorem 8 (Kato-Temple bound)" + Let $\opA$ be a self-adjoint operator, + $\tilde \varphi \in D(\opA)$ with $\| \tilde \varphi \| = 1$, $\tilde \lambda = q_A(\tilde \varphi)$, + and $r = \opA \tilde \varphi - \tilde \lambda \tilde \varphi$. Let $\lambda$ be + the eigenvalue closest to $\tilde \lambda$ and + ```math + \begin{align} + \delta = \min_{s \in \sigma(\opA) \setminus \{ \lambda \} } | s - \tilde \lambda | + + \end{align} + ``` + such that + $\sigma(\opA) \cap [ \tilde \lambda - \delta, \tilde \lambda + \delta ] = \{ \lambda \}$. + Then + ```math + \begin{align} + | \tilde \lambda - \lambda | \leq \frac{\| r \|^2}{\delta} + + \end{align} + ``` +""" + +# ╔═║ 3bee37f2-ec13-4799-b61e-af6879e3b7cc +md""" + +Note that this results follows automatically if we can prove the following : + +!!! note "Theorem 9 (Temple's Inequality)" + Let + $\opA, \tilde \varphi, \tilde \lambda, r$ as in + Theorem 1. Suppose $\alpha, \beta \in \mathbb R$ with + $\alpha < \tilde \lambda < \beta$ and + $(\alpha, \beta ) \cap \sigma(\opA) = \{\lambda \}$. Then + ```math + \begin{align} + \tilde \lambda - \frac{\| r\|^2}{\beta - \tilde \lambda} \leq \lambda \leq \tilde \lambda + \frac{\| r \|^2}{\alpha - \tilde \lambda} + + \end{align} + ``` +""" + +# ╔═║ 5731ef65-9c89-4859-b891-f5c2606c5772 +md""" + +To prove this, we need a bit of spectral calculus. +Like in the finite-dimensional case, we have : + +!!! note "Theorem 10 (Cauchy's Formula)" + Let $\opA$ with + $D(\opA) \leq \hilbert$ be a self-adjoint operator. Let $a,b \in \resolvent(\opA) \cap \mathbb R$ + with $a=latex,->] (-5,0) -- (5,0) node[above right,blue]{$\sigma(A)$} ; + \draw (5,0) node[right]{$\mathbb{R}$}; + + + %sigma_p + + + \foreach \energy in {-4,-1,2,2.5,3.5}{ + \draw[blue] (\energy,0) node{$\times$}; + } + + \draw[ultra thick,color=blue] (-3,0) -- (-2,0); + \draw[ultra thick,color=blue] (0,0) -- (1.5,0); + + \draw[rounded corners,purple] (-0.5,0.5) rectangle (2.25,-0.5) {}; + + + \draw[purple] -- (0.875,0.5) node{$<$} -- (2.25,0.5) -- (2.25,-0.5) node[below]{$b$} -- (0.875,-0.5) node{$>$} -- (-0.5,-0.5) node[below]{$a$}; + +""",width="20cm",options="scale=1",preamble=raw"\usepackage{amsfonts}") + +# ╔═║ 8c52943d-a2d6-4fa7-bae5-2c7c6fe54140 +md""" + +Illustration of Cauchy's formula. + +--- + +!!! tip "Remarks" + * $a,b \notin \sigma(\opA)$ + * Similar to our discussion in the finite-dimensional case $\mathbf 1_{[a,b]} (\opA)$ is a projector. + + +Of special importance are the *spectral projectors* +```math +\begin{align} + P^\opA (\lambda) = \mathbf 1 _{(-\infty , \lambda ]} (\opA) +\end{align}. +``` + +To gather some understanding, we first study the finite-dimensional case : + +""" + +# ╔═║ 06fc9dd1-601c-4d4c-9f86-ddb9be7fc13f +md""" +##### Finite dimensions + +We study the finite dimensional case with a matrix +$M \in \mathbb R^{d \times d}$ with distinct eigenvalues +$\lambda_1 < \dots < \lambda_m ( m \leq d)$. +Note that we assume single eigenvalues for simplicity. +In this setting +```math +\begin{align} + P^M(\lambda) = \mathbf 1 _{(-\infty,\lambda]} (M) = \bigoplus_{\lambda_i \leq \lambda} \mathop{\mathrm{Ker}}(M - \lambda_i) +\end{align} +``` +where $\text{Ker}(M - \lambda_i)$ is th eigenspace of eigenvalue $\lambda_i$. +We notice that $P^M (\lambda)$ is piecewise constant, and $P^M (\lambda) \in \mathbb R^{d \times d}$. +Therefore, we can compute +its derivatives with respect to $\lambda$ (in the distributional sense) : +```math +\begin{align} + \frac{dP^M(\lambda)}{d \lambda} = \sum_{i=1}^d \delta (\lambda_i - \lambda_j) \mathbf 1 _{\{ \lambda_j \}} (M) +\end{align} +``` +where $\mathbf 1 _{\{ \lambda_j \}} (M)$ is the eigenspace of eigenvalue $\lambda_j$. +By integration, we obtain in particular +```math +\int_\mathbb R \lambda dP^M(\lambda) = \int_\mathbb R \lambda \frac{dP^M(\lambda)}{d \lambda} d\lambda = \sum_{i=1}^d \lambda_j \mathbf 1 _{\{ \lambda_j \}} (M) = M. +``` + +Now, we want to write something similar for $\opA$ in infinite dimensions. +""" + +# ╔═║ 66c953ac-d572-469a-94f1-67cc473c5704 +md""" +##### Infinite dimensions + +We fix $v \in D(\opA)$. +Now the function +```math +\begin{align} + P_v(\lambda) = \langle v, \mathbf 1 _{(-\infty, \lambda]} (\opA) v \rangle = \langle v, P^\opA(\lambda) v \rangle +\end{align} +``` +turns out to be bounded and increasing, making its +distributional derivative a measure, called the *spectral measure*. With +this, we can write +```math +\begin{align} + \langle v, Av \rangle = \int_\mathbb R \lambda \ d \langle v, P^\opA (\lambda) v \rangle. +\end{align} +``` +which is the infinite dimensional form of spectral resolution. +More generally, for any (measurable) function $f$ +and $v,w \in D(\opA)$ +```math +\begin{align} + \langle v , f(\opA) w \rangle = \int_\mathbb R f(\lambda) \ d \langle v, P^\opA (\lambda) v \rangle. +\end{align} +``` +which can in turn be used to define a self-adjoint operator $f(\opA)$ (functional calculus). +Without going into details with this the following result is plausible : +""" + +# ╔═║ 2de385c1-8ad1-4cbf-a997-473daf2bcc33 +md""" +!!! note "Lemma 11" + If $f$ is a polynomial and $\opA$ is self-adjoint, then + ```math + \sigma(f(\opA)) = \{ f(\lambda) \mid \lambda \in \sigma(\opA) \}. + ``` + +With this result, we can finally prove +""" + +# ╔═║ 85c4f6c2-23dc-45b8-b733-f267b03cdde5 +md""" +> **Proof of Theorem 9.** +> We note that our assumptions imply $(\lambda , \beta) \cap \sigma(\opA) = \varnothing$ such that, using Lemma 11 and the polynomial $f(x) = (x - \beta) (x - \lambda)$, we obtain that $(\opA - \beta) (\opA - \lambda) \geq 0$, i.e. that the operator $(\opA - \beta) (\opA - \lambda)$ only has non-negative spectrum. +> Therefore +> ```math +> \begin{align} +> 0 &\geq \langle \tilde \varphi, (\opA - \beta) (\opA - \lambda) \tilde \varphi \rangle +> \\ +> &= \langle \tilde \varphi, (A - \tilde \lambda + \tilde \lambda - \beta) (\opA - \tilde \lambda + \tilde \lambda - \lambda) \tilde \varphi \rangle +> \\ +> &= \| r \|^2 + \langle \tilde \varphi, (\opA - \tilde Ξ») \tilde \varphi \rangle (\tilde Ξ» - Ξ») +> \\ +> & \qquad + (\tilde Ξ» + \beta ) \langle \tilde \varphi , (\opA - \tilde Ξ») \tilde \varphi \rangle +> \\ +> & \qquad + (\tilde Ξ» - \beta ) (\tilde Ξ» - Ξ») +> \\ +> & = \| r \|^2 + (\tilde Ξ» - \beta ) (\tilde Ξ» - Ξ» ) +> \end{align} +> ``` +> Dividing by $(\tilde Ξ» - \beta)$ and rearranging yields the first inequality. +> Similarly, $(\opA - \alpha) (\opA -\lambda) \geq 0$ yields the second. +> $\hspace{8cm} \square$ +""" + +# ╔═║ 4d4d6fa7-99c5-4cb7-8d3e-830634a37daf + + +# ╔═║ e1d81761-c3c4-4bef-9e14-96bdcf5c8eba +TableOfContents() + +# ╔═║ 5c5e2ead-9487-42a4-b049-ce7b90cc97d2 +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 305) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +QuadGK = "1fd47b50-473d-5c70-9696-f719f8f3bcdc" +TikzPictures = "37f6aa50-8035-52d0-81c2-5a1d08754b2d" + 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\ No newline at end of file diff --git a/11_Periodic_problems.jl b/11_Periodic_problems.jl new file mode 100644 index 0000000..0d95c2c --- /dev/null +++ b/11_Periodic_problems.jl @@ -0,0 +1,1343 @@ +### A Pluto.jl notebook ### +# v0.19.43 + +using Markdown +using InteractiveUtils + +# ╔═║ a0407e4d-3edc-48ea-a335-b815b796e378 +begin + import TikzPictures.TikzPicture + using LaTeXStrings + using PlutoUI + using PlutoTeachingTools + using HypertextLiteral + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ e747d030-7487-4598-99d6-93afefd425e1 +md""" +# Periodic Problems +""" + +# ╔═║ 1441bdfc-715e-46da-a4d2-8b788c71853f +md""" +In the previous lecture we saw that the discrete eigenvalues below the +essential spectrum of an operator can be approximated. +Indeed, if $(\lambda_k, \phi_k)$ denotes the $k$-th such eigenpair (below $\Sigma(\mathcal A)$), then we can choose a *discrete basis* $\{ \phi_i \}_i$ spanning a subspace $S \leq D (\mathcal A)$ and +find an approximation $(\tilde \lambda_k , \tilde \phi_k)$ as the $k$-th +eigenpair of the matrix +```math +\begin{align} + A_{ij} \equiv \langle \phi_i | \mathcal A \phi_j \rangle. +\end{align} +``` + +Due to Courant-Fisher the approximation is from above and due to +Kato-Temple the error is controlled. + +As an example for understanding this discretization procedure we will +discuss this in chapter the special setting of periodic problems with +Hamiltonian $\opH = - \frac1{2} \laplacian + V$, where $V$ is periodic. +Albeit such Hamiltonians have *no eigenvalues*, the Bloch-Floquet +transform will still allow us to construct a matching basis and +approximate the spectra of these operators in a meaningful way. +""" + +# ╔═║ 7971ffe3-39f0-4cf8-9cee-fd55483a8ed9 +md""" +## Periodic functions and lattices + +A periodic problem is characterized by being invariant to certain translations. + +- For example, the sine function satisfies + ```math + \begin{align} + \sin(x) = \sin(x + 2 \pi m) && \forall m \in \mathbb Z + \end{align} + ``` + making it $2 \pi$-periodic. Alternatively, we could denote this as + ```math + \begin{align} + \sin(x) = \sin(x+R) && \forall R \in \mathbb L + \end{align} + ``` + where $\mathbb L$ is the *lattice*, $\mathbb L = \{ 2 \pi m \mid m \in \mathbb Z \} = 2 \pi \mathbb Z$. + + +- The importance of periodicity for numerical computations is as follows : once an $\mathbb L$-periodic function, like $\sin$, is known inside the *unit cell* $\Omega$, the open and bounded set + ```math + \begin{align} + \Omega = \{ x \in \mathbb R \mid |x-R| > |x| \quad \forall R \in \mathbb L \setminus \{ 0 \} \} + \end{align} + ``` + it is actually known for everywhere in $\mathbb R$. + This suggests that solving periodic problems inside the unit cell should be sufficient to know the solution everywhere. + In the case of $\sin$, the unit cell is $(-\pi,\pi)$. +""" + +# ╔═║ 8d9fe76f-dfb3-4683-83c2-5d7cd2aff712 +TikzPicture(L""" + \draw (-5,0) node[left]{$\mathbb L$} + -- (-4,0) node{$\bullet$} node[below]{$-4 \pi$} + -- (-2,0) node{$\bullet$} node[below]{$-2 \pi$} + -- (-1,0) node[above]{$- \pi$} + -- (0,0) node{$\bullet$} node[below]{$0$} + -- (1,0) node[above]{$\pi$} + -- (2,0) node{$\bullet$} node[below]{$2 \pi$} + -- (4,0) node{$\bullet$} node[below]{$4 \pi$} + -- (5,0) + ; + + \draw (-6.5,0) ; + + \draw (-1,0.5) node{$ ]$} -- (0,0.5) node[above]{$\Omega$} -- (1,0.5) node{$[$} ; +""",width="23cm",options="scale=1",preamble=raw"\usepackage{amsfonts}") + +# ╔═║ df23c23e-b509-4f48-9ab9-dd2b59834d01 +md""" +- These concepts naturally generalize to higher dimensions, e.g. a 3D lattice is + ```math + \begin{align} + \mathbb L = \{ \mathbb Z a_1 + \mathbb Z a_2 + \mathbb Z a_3 \} + \end{align} + ``` + where $a_1,a_2,a_3 \in \mathbb R ^3$ are three linearly independent vectors. They define a unit cell + ```math + \begin{align} + \Omega = \{ x \in \mathbb R^3 \mid |x-R| > |x| \quad \forall R \in \mathbb L \setminus \{ 0 \} \}. + \end{align} + ``` + + Frequently, the matrix + ```math + \begin{align} + A = \begin{pmatrix} + a_1 & a_2 & a_3\\ + \downarrow & \downarrow & \downarrow + \end{pmatrix} + \in \mathbb R ^{3 \times 3} + \end{align} + ``` + is considered. + +""" + +# ╔═║ 9b17dde4-6652-4800-a568-1244cf519593 +md""" +- A function $f$ is now called $\mathbb L$-periodic if + ```math + \begin{align} + f(x) = f(x + R) && \forall R \in \mathbb L + \end{align} + ``` + and almost every $x \in \mathbb R^3$. + + +- Based on this observation we introduce the translation operator + ```math + \begin{align} + \mathcal T_R : C_0^\infty (\mathbb R^3) \to C_0^\infty (\mathbb R^3), f \mapsto f(\bullet - R) + \end{align} + ``` + which by denseness of $C_0^\infty$ on $L^2$ can be extended to all $L^2$ functions. +""" + +# ╔═║ fbbe239e-c06e-41b1-b1e3-857585f994bd +md""" +- Further we introduce the function space + ```math + \begin{align} + L^2_{per} (\Omega) = \{ f \in L^2_{loc} (\mathbb R^3) \mid f \text{ is } \mathbb L \text{ periodic} \} + \end{align} + ``` + with inner product + ```math + \begin{align} + \langle f, g \rangle _{L_{per}^2 (\Omega)} = \int_{\Omega} \overline{f(x)} g(x) dx. + \end{align} + ``` + + +- A convenient basis for $L^2_{per} (\Omega)$ is the plane wave or Fourier basis + ```math + \begin{align} + \mathbb B = \left \{ e_G(x) \equiv \frac1{\sqrt{\Omega}} e^{i G x} \middle \vert G \in \mathbb L ^* \right \} + \end{align} + ``` + where $\mathbb L^*$ denotes the *reciprocal lattice* + + ```math + \begin{align} + \mathbb L^* = \{ \mathbb Z b_1 + \mathbb Z b_2 + \mathbb Z b_3 \} + \end{align} + ``` + such that + ```math + \begin{align} + a_i \cdot b_j = 2 \pi \delta_{ij} && \text{for } i,j \in \{1,2,3\} + \end{align} + ``` + where $\delta_{ij}$ is the Kronecker delta, i.e. + ```math + B = 2 \pi A^{-1} + ``` + with $B = (b_1,b_2,b_3)$. + +""" + +# ╔═║ ea689cad-2859-41b0-aef3-36651a8d166b +md""" +- Note that the construction of $\mathbb L^*$ is such to ensure that $e_G(x)$ is $\mathbb L$-periodic by enforcing + ```math + \begin{align} + e^{i R \cdot G} = 1 && \forall \ R \in \mathbb L, G \in \mathbb L^* + \end{align} + ``` + + +- We also define $\Omega^*$ as the unit cell of $\mathbb L^*$, also called the *first Brillouin zone*. +""" + +# ╔═║ 6ba1ce9a-6ced-4a82-9b5a-6b07a25c534b +md""" +- Finally, we introduce some concepts from Fourier analysis. + For a function $f \in L^2_{per} (\Omega)$ the following expressions are frequently encountered : + ```math + \begin{align} + f = \sum_{G \in L^*} \hat f_G e_G \tag{Fourier Series Expansion} + \end{align} + ``` + with + ```math + \begin{align} + \hat f_G \equiv \langle f | e_G \rangle = \int_{\Omega} f e^{- i G x} dx \tag{Fourier Coefficients} + \end{align} + ``` + and +```math +\begin{align} + \| f \|_{L^2_{per} (\Omega)} = \int_\Omega | f |^2 dx = \sum_{G \in \mathbb L^*} |\hat f_G|^2 < \infty. \tag{Parseval's Identity} +\end{align} +``` + +""" + +# ╔═║ 6c5c1231-9e16-4a43-8dec-68df585d78b9 +md""" +- Employing a Fourier basis, *periodic Sobolev spaces* can be easily characterized as + ```math + \begin{align} + H^S_{per} (\Omega) = \left \{ f \in L^2_{per} (\Omega) \middle | \sum_{G \in \mathbb L^*} (1 + |G|^2)^S + |\hat f_G|^2 < \infty \right \} + \end{align} + ``` + where $S>0$. +""" + +# ╔═║ 32086662-d60d-47cd-acb6-0f1a4f18e351 +md""" +## Periodic operators and periodic schrΓΆdinger problems + +!!! note "Definition (L-periodicity)" + A linear operator $\mathcal A : L^2(\mathbb R^d) \to L^2(\mathbb R^d)$ is called + $\mathbb L$-periodic if it commutes with translations + $\mathcal T_R$ for all $R \in \mathbb L$, i.e. + ```math + \begin{align} + \mathcal A (\mathcal T_R (\varphi)) = \mathcal T_R(\mathcal A (\varphi)) && + \forall \ R \in \mathbb L, \varphi \in L^2(\mathbb R^d). + + \end{align} + ``` + + + +!!! note "Theorem 1" + Let $V \in L^{3/2}_{per} (\Omega)$ for $\Omega \in \mathbb R^3$. The + operator + ```math + \opH = - \frac1{2} \laplacian + V + ``` + is self-adjoint on + $\hilbert = L^2(\mathbb R^3)$ with $D(\opH) = H^2(\mathbb R^3)$ and $Q(\opH) = H^1(\mathbb R^3)$. + $\opH$ is also bounded from below. + + +A perhaps surprising result is that, while $\opH$ is $\mathbb L$-periodic, +not all eigenfunctions of $\opH$ are $\mathbb L$-periodic. + + +""" + +# ╔═║ 3a8ff44a-c150-483a-b2f1-4fa085dd9075 +md""" +## Bloch-Floquet theory + +Since $\widehat{\laplacian + f} (p) = - |p|^2 \hat f (p)$, the Fourier +transform +```math +\begin{align} + \hat f(p) = \frac1{(2 \pi)^{\frac{3}{2}}} \int_{\mathbb R} f(x) e^{-i p \cdot x} dx +\end{align} +``` + provides exactly the unitary transformation required to +diagonalize the Laplacian $\laplacian$. Notably eigenfunctions are the +plane wave $e^{i p \cdot x}$ with $p \in \mathbb R^3$. +Note that other Fourier transform conventions exist. +Here we explicitly choose the one which is unitary (which is *not* the +one used in DFTK). + +In our problem $\opH = - \frac1{2} \laplacian+ V$ with $V$ +$\mathbb L$-periodic. We thus need an equivalent operation, that is +better adapted to the $\mathbb L$-periodic setting. This turns out to be +the *Bloch transform*. + +""" + +# ╔═║ 083e1f7a-e09e-479a-9cd9-83203e8c23dc +md""" +- First, we rewrite the plane waves as + ```math + \{ e^{i (k+G) \cdot x} \} _{k \in \Omega^*, G \in \mathbb L^*} + ``` + where we note that for all $p \in \mathbb R^3$ a decomposition $p = k+G$ with $G \in \mathbb L^*, k \in \overline{\Omega^*}$ is always possible. + + +- Then, we use this to perform the inverse transform in two steps + ```math + \begin{align} + f(x) &= \frac1{(2 \pi)^{\frac{3}{2}}} \int_{\mathbb R^3} \hat f (p) e^{i p \cdot x} + \\ + &= \frac1{(2 \pi)^{\frac{3}{2}}} \sum_{G \in \mathbb L^*} \int_{\Omega^*} \hat f(G+k) e^{i (k+G)\cdot x} dk + \end{align} + ``` + where we take an $f \in C_0^\infty (\mathbb R^3)$. + By density arguments our discussion can be easily extended to $L^2 (\mathbb R^3)$. + +""" + +# ╔═║ f8c35c68-0e60-45d7-ab93-5aace2bd1589 +md""" +- From this, we introduce the **Bloch transform** $\bloch$ + ```math + \begin{align} + (\bloch f) (x) = f_k(x) &= \frac1{\sqrt{|\Omega|}} \sum_{G \in \mathbb L^*} \hat f (G+k) e^{i G\cdot x} + \\ + &= \sum_{G \in \mathbb L^*} \hat f (k+G) e_G(x). + \end{align} + ``` + We arrive at the reconstruction formula by considering the second step of the inverse Fourier transform + ```math + \begin{align} + f(x) = \frac1{\sqrt{|\Omega^*|}} \int_{\Omega^*} f_k(x) e^{i k\cdot x} dk + \end{align} + ``` + where we used that + ```math + \begin{align} + \frac1{\sqrt{|\Omega|}} = \frac{\sqrt{|\Omega^*|}}{(2 \pi)^{\frac{3}{2}}}. + \end{align} + ``` + Notice that this normalization convention is chosen such that the normalized plane waves appear in the Bloch transform. + +""" + +# ╔═║ db338525-379e-44e5-b3eb-8d98e33d171b +md""" +- The Bloch transform satisfies + + ```math + \begin{align} + f_k(x+R) &= f_k(x) \tag{Periodicity} + \\ + f_{k+G} (x) &= e^{- i G \cdot x} f_k(x). \tag{Born-von Karman condition} + \end{align} + ``` + +- As a result we can always restrict $f_k(x)$ to $k \in \Omega^*$ and $x \in \Omega$. + +""" + +# ╔═║ 9a6793e3-a46a-4736-9521-c5c6a1636aa3 +md""" +- An explicit construction of $f_k$ from $f$ is available via **Poisson's formula** : + ```math + \begin{align} + f_k(x) &= \frac1{\sqrt{|\Omega|}} \sum_{G \in \mathbb L^*} \hat f (G + k) e^{i G \cdot x} + \\ + &= \frac1{\sqrt{|\Omega^*|}} \sum_{R \in \mathbb L} f(x+R) e^{- i k \cdot (x +R)} + \end{align} + ``` + This can be seen from + ```math + \begin{align} + \hat f(G+k) &= \frac1{(2 \pi)^{\frac{3}{2}}} \int_{\mathbb R^3} f(x) e^{-i (G + k) \cdot x} dx + \\ + &= \frac1{(2 \pi)^{\frac{3}{2}}} \sum_{R \in \mathbb L} \int_\Omega f(R+y) e^{-i (G+k) \cdot (R+y)} dy + \\ + &= \frac1{\sqrt{|\Omega^*|}} \int_\Omega \left ( \sum_{R \in \mathbb L} f(R+y) e^{-i k \cdot (R+y)} \right ) \frac{e^{-i G \cdot y}}{\sqrt{|\Omega|}} dy + \end{align} + ``` + and the fact that + ```math + \begin{align} + \sum_{G \in \mathbb L^*} e^{i G \cdot (x-y)} = \delta (x-y). + \end{align} + ``` + From Poisson's formula it is clear that *Parceval's identity* + ```math + \int_{\mathbb R^3} | f(x) |^2 dx = \int_{\Omega^*} dk \int_\Omega dy \ |f_k(y)|^2 + ``` + holds, since + ```math + \begin{align} + \int_{\Omega^*} dk \int_\Omega dy \ |f_k(y)|^2 &= \frac1{|\Omega^*|} \sum_{R, R' \in \mathbb L} \left ( \int_{\Omega^*} e^{i k \cdot (R-R')} dk \right ) \left ( \int_\Omega \overline{f(y+R')} f(y+R) dy \right ) + \\ + &= \sum_{R \in \mathbb L} \int_\Omega |f(y+R)|^2 \ dy + \\ + &= \int_{\mathbb R^3} |f(x)|^2 \ dx + \end{align} + ``` +""" + +# ╔═║ 17d99340-383a-4b29-a1a2-244024c71155 +md""" +We summarize our findings in a Theorem : + +!!! note "Theorem 2 (Bloch-Floquet transform)" + The Bloch-Floquet transform $\bloch$ is the unitary mapping from + $L^2(\mathbb R^d)$ to the vector space + $L^2_{qp} (\Omega^*, L^2_{per}(\Omega))$ (defined below) + with the property that each $f \in C_0^\infty (\mathbb R^d)$ is mapped to + ```math + \Omega^* \ni K \mapsto f_k = \sum_{R \in \mathbb L } u(\bullet + R) e^{- k \cdot (\bullet + R)} \in L^2_{per}(\Omega). + ``` + The extension to $L^2(\mathbb R^d)$ follows by density of $C_0^\infty$ in $L^2$. + + In this $L^2_{qp} (\Omega^*, L^2_{per}(\Omega))$ is the space of $\mathbb L^*$-*quasiperiodic* functions which are $L^2_{per}(\Omega)$-valued : + ```math + \begin{align} + L^2_{qp} (\Omega^*, L^2_{per}(\Omega)) = + \left \{ \mathbb R^d \ni k \mapsto u_k \in L^2_{per}(\Omega) \middle \vert \int_{\Omega^*} \| u_k \|^2_{L^2_{per}(\Omega)} \ dk < \infty \right . + \\ + \text{where } u_{k+G} = u_k e^{-i G \cdot x} \text{ for all } G \in\mathbb L^* + \\ + \text{and almost every } k \in \mathbb R^d \bigg \}. + + \end{align} + ``` + On this vector space we have the inner product + ```math + \langle f,g \rangle_{L^2_{qp} (\Omega^*, L^2_{per}(\Omega))} = \frac1{|\Omega^*|} \int_{\Omega^*} \langle f_k, g_k \rangle_{L^2_{per}(\Omega)} dk. + ``` +""" + +# ╔═║ fcf275be-b4c2-4d06-b502-a0c52e74b533 +md""" +!!! tip "Remark" + By way of Parceval's identity we can also understand $L^2_{qp} (\Omega^*, L^2_{per}(\Omega)) \equiv L^2(\Omega^* \times \Omega)$, i.e. $\bloch$ maps single variable functions of $y \in \mathbb R^d$ to functions of two variables : $k \in \Omega^*$ and $x \in \Omega$. + + We can illustrate it as + ```math + \begin{align} + L^2(\mathbb R^d) && \stackrel{\bloch}{\longrightarrow} && L^2_{qp} (\Omega^*, L^2_{per}(\Omega)) = L^2(\Omega^* \times \Omega) + \\ + \mathbb R^d \ni x \mapsto f(x) \in \mathbb C &&&& + \Omega^* \times \Omega \ni (k,y) \mapsto f_k(y) \in \mathbb C + \end{align} + ``` +""" + +# ╔═║ 9d4a8178-52f4-4846-94d6-74b8d68abbb4 +# use y instead of x?? +md""" +In physics texts it is frequently written that the Bloch transform admits the decomposition $\psi(x) = u_{k'}(x) e^{i k' \cdot x}$ for an eigenstate $\psi$, where $u_k$ is $\mathbb L$-periodic. We want to make the connection of this with the present discussion. +As we will see in the following, an eigenvalue can be restricted to a single value of $k$. + +Thus, we define $ψ_k(x)$ as +```math +k \mapsto \begin{cases} u_{k'}(x) e^{- i G \cdot x} & \text{if } k = k'+G \text{ for } G \in \mathbb L \\ 0 & \text{otherwise} \end{cases} +``` +The reconstruction formula then gives +```math +\begin{align} + ψ(x) &= \frac1{\sqrt{|\Omega^*|}} \int_{\Omega^*} ψ_k(x) e^{ik \cdot x} dk + \\ &= \frac1{\sqrt{|\Omega^*|}} u_{k'}(x) e^{i k' \cdot x} +\end{align} +``` +which is just a scaled form of the commonly quoted representation. +""" + +# ╔═║ 21523a64-b1a9-422d-8dc0-ebb324f35380 +#check the formula +md""" +The next natural question is to consider the Bloch-Floquet transform of operators. +We first consider the Laplace operator. +Using Poisson's formula, +```math + f_k(x) = \frac1{\sqrt{|\Omega^*|}} \sum_{R \in \mathbb L} f(x + R) e^{- i k \cdot (x+R)} +``` +we obtain +```math +\begin{align} + (-i \nabla_x + k) f_k(x) &= \frac1{\sqrt{|\Omega^*|}} \sum_{R \in \mathbb L} [- i \nabla_x f(x+R)] e^{- ik \cdot (x+R)} + + \\ &\quad + f(x+R) e^{-i k \cdot (x+R)} [(-i)^2 k + k] + \\ &= \frac1{\sqrt{|\Omega^*|}} \sum_{R \in \mathbb L} [- i βˆ‡_x f(x+R)] e^{-i k (x+R)} + \\ &= [- i βˆ‡_x f]_k (x). +\end{align} +``` +This is clearly true if $f \in C_0^\infty (\mathbb R^3)$. +Again by density arguments this shows +```math +\bloch H^1(\mathbb R^3) = L^2_{qp} (\Omega^*, H^1_{per}(\Omega)). +``` +Similarly +```math +\bloch (- \laplacian _x) f = (- \laplacian_x f)_k = \lvert -i \nabla_k + k |^2 f_k = \lvert -i \nabla_k + k |^2 \bloch f +``` +suggesting +```math +\bloch H^2(\mathbb R^3) = L^2_{qp} (\Omega^*, H^2_{per} (\Omega)) +``` +i.e. that regularity in $f$ induces equal regularity in $f_k$ for all $k \in \Omega^*$. + +Notably, this also suggests the Bloch transform of the kinetic energy operator $- \laplacian_x$ as +```math +\bloch (- \laplacian_x) \bloch^{-1} = \lvert - i βˆ‡_x + k|^2 +``` +interpreted in the sense of distributions and defined on the Hilbert space $L^2_{qp} (\Omega^*, L^2_{per} (\Omega))$ with domain $D( \bloch (- \laplacian_x) \bloch^{-1} ) = L^2_{qp} (\Omega^*, H^2_{per} (\Omega))$. + +This result is formalized in the following theorem. +""" + +# ╔═║ 62e5c394-dad2-4a62-a7ae-d4999f741505 +md""" +Denoting $\mathscr L (V) = \{ f : V \to V | f \text{ linear} \}$ the space of linear maps between a function space and itself, we have : + +!!! note "Theorem 3 (Bloch fibers)" + Any bounded $\mathbb L$-periodic operator $\opA$ on $L^2(\mathbb R^3)$ is decomposed by the Bloch-Floquet transform in the following sense : there exists a function $k \mapsto \opA_k$ in $L^\infty_{qp} (\mathbb R^3, \mathscr L(L^2_{per} (\Omega))$ such that for any $u \in L^2(\mathbb R^3)$, any $G \in \mathbb L^*$, and almost any $k \in \Omega^*$ is holds that + ```math + \begin{align} + (\opA u)_k = \opA_k u_k && \opA_{k+G} = e^{-i G \cdot x} \opA_k e^{i G \cdot x} + \end{align} + ``` + where the operators $(\opA_k)_{k \in \Omega^*}$ on $\hilbert = L^2_{per} (\Omega)$ are called the *Bloch fibers* of $\opA$. + + The Bloch decomposition can be extended to unbounded operators on $L^2(\mathbb R^3)$, in which case $\opA_k$ are unbounded as well. + +!!! warning "Example 1 (Laplace operator)" + For $\opA = - \laplacian$, the Bloch fibers aro $\opA_k = \lvert - i \nabla_x + k|^2$. +""" + +# ╔═║ 6da749bc-0181-49c2-8a7a-9c180930dce6 +md""" +## Fibers of periodic SchrΓΆdinger operators + +We consider $\opH = - \frac1{2} \laplacian + V$ on $L^2(\mathbb R^3)$ with domain $D(\opH) = H^2(\mathbb R^3)$ and $V \in L^2_{per} (\mathbb R^3)$ such that $\opH$ is $\mathbb L$-periodic. +Recall this operator is self-adjoint and as *no* eigenvalues. + +- The fibers of $\opH$ are $\opH_k = \lvert - i \nabla_x + k|^2 + V$, where we note + - $\opH_k$ is a self-adjoint and bounded from below operator on $L^2_{per}(\Omega)$ with domain $H^2_{per}(\Omega)$ and form domain $H^1_{per}(\Omega)$ + - $\opH_k$ has a compact resolvent $\resolvent(\opH_k)$, such that each $\opH_k$ has a purely discrete spectrum with eigenvalues accumulating at $+ \infty$ and eigenfunctions forming an orthonormal basis for $L^2_{per}(\Omega)$. + + +- To understand the spectrum of $\opH$, it suffices completely to study the spectrum of all $\opH_k$. + Indeed + ```math + \sigma(\opH) = \bigcup_{k \in \overline{\Omega^*}} \sigma(\opH_k). + ``` + + +- We thus seek an $L^2_{per} (\Omega)$ orthonormal basis $\{ (Ξ»_{k,n}, \phi_{k,n} )\}_{n \in \mathbb N} \subset (\mathbb R \times L^2_{per}(\Omega))^{\mathbb N}$ of eigenstates of $\opH_k$ : + ```math + \begin{align} + \opH_k \phi_{k,n} &= Ξ»_{k,n} \phi_{k,n} + \\ + \langle \phi_{k,n} , \phi_{k,m} \rangle_{L^2_{per}(\Omega)} &= \delta_{mn} && \forall n,m \in \mathbb N. + \end{align} + ``` + + By convention, we sort for each $k \in \overline{\Omega^*}$ : $Ξ»_{k,1} ≀ Ξ»_{k,2} ≀ Ξ»_{k,3} ≀ …$, thus counting multiplicities. + Moreover, for each $n \in \mathbb N$ we define the mapping + ```math + Ξ»_n : \overline{\Omega^*} \to \mathbb R : k ↦ Ξ»_{k,n} + ``` + which is a Lipshitz function called the $n$-th **energy band**. + + +- Each band has a well-defined minimum/maximum, forming the following interval + ```math + Ξ»_n \left (\overline{\Omega^*} \right ) = \left [ \min_{k ∈ \overline{\Omega^*}} Ξ»_n(k) , \max_{k ∈ \overline{\Omega^*}} Ξ»_n(k) \right ]. + ``` + Moreover, + ```math + Οƒ(\opH) = \bigcup_{n ∈ \mathbb N} Ξ»_n \left ( \overline{\Omega^*} \right ) + ``` + +--- + +An image to have in the head is +""" + +# ╔═║ 08b20ea7-f3c3-4302-bfc3-a13bbd57930a +TikzPicture(L""" + %connecting dashes + \draw[dashed,blue] (220,230) -- (250,230) -- (250,200) -- (189.5,200); + \draw[dashed,cyan] (220,190) -- (249.6,190) -- (249.6,160) -- (190.5,160); + \draw[dashed,teal] (125.5,170) -- (250.4,170) -- (250.4,130) -- (220,130); + \draw[dashed,olive] (165.5,100) -- (250,100) -- (250,80) -- (190,80); + + + %1st band + \draw[blue,thick] + (100,230) .. controls (108.74,223.57) and (104.5,213.5) .. + (127.5,211.5) .. controls (150.5,209.5) and (175.02,200.09) .. + (189.5,200) node[above]{$\lambda_1$} .. controls (203.98,199.91) and (211,210) .. + (220,230) ; + + %2nd band + \draw[cyan,thick] + (100,190) .. controls (106.84,188.97) and (135.14,187.24) .. + (157.5,179) .. controls (179.86,170.76) and (176.5,160) .. + (190.5,160) node[above]{$\lambda_2$} .. controls (204.5,160) and (200.5,178) .. + (220,190) ; + + %3rd band + \draw[teal,thick] (100,130) .. controls (114.5,130) and (111.5,170) .. + (125.5,170) .. controls (139.5,170) and (135.5,140) .. + (149.5,140) node[above]{$\lambda_3$} .. controls (163.5,140) and (166.5,162) .. + (175.5,146) .. controls (184.5,130) and (212.5,143) .. + (220,130) ; + + %4th band + \draw[olive,thick] + (100,90) .. controls (115.5,84) and (107.5,94) .. + (114.5,94) .. controls (121.5,94) and (119.5,81) .. + (129.5,83) .. controls (139.5,83) and (135.5,87) .. + (143.5,91) node[below]{$\lambda_4$} .. controls (151.5,98) and (146.55,83.87) .. + (155.5,85) .. controls (164.45,86.13) and (155.5,101) .. + (165.5,100) .. controls (175.5,101) and (172.5,81) .. + (190,80) .. controls (208.5,81) and (205.5,105) .. + (220,90) ; + + %axes + \draw[<-,>=latex] (100,70) node[left]{$\lambda$} -- (100,240) -- (165,240) node[below]{$k \in \Omega^*$} -- (220,240) -- (220,70) ; + + %spectrum + \draw[<-,>=latex] (250,70) node[right]{$\sigma(- \Delta/2 + V)$} -- (250,240) ; + + \draw[very thick,blue] (250,230) -- (250,200); + \draw[very thick,cyan] (249.6,190) -- (249.6,160); + \draw[very thick,teal] (250.4,170) -- (250.4,130); + \draw[very thick,olive] (250,100) -- (250,80); + + %band gaps + \draw[<->] (260,100) -- (260,115) node[right]{Bandgap} -- (260,130); + \draw[<->] (260,200) -- (260,195) node[right]{Bandgap} -- (260,190); + + + + +""",width="17cm",options="xscale=0.05,yscale=-0.05",preamble=raw"\usepackage{amsfonts}") + +# ╔═║ 41c0f837-055c-4077-b7f8-966e84ee709f +md""" +--- + +## Numerical treatment of periodic treatments + +From our treatment it is clear that the lower end of the spectrum of $\sigma(\opH)$ for periodic potentials can be approximated by considering the lowest few bands via computing the lowest few eigenpairs of the fibers $\opH_k, k \in \overline{\Omega^*}$. + +- For each $k \in \overline{\Omega^*}$, $\opH_k$ has only eigenvalues. + Projection techniques are thus well-suited to approximate its eigenpairs from above. + Moreover, Temple's inequality can be used for an a posteriori error bound. + + +- In practice, we select first a subset $\mathbb K \subset \overline{\Omega^*}$, the so-called *$k$-grid* or *$k$-point mesh*. + + +- A popular basis for discretizing $\opH_k$ and solving for its lowest-energy eigenstates using a Ritz-Galerkin projection method are the *k-adapted plane waves* + ```math + \mathbb B^{E_{crit}}_{k} = \left \{ e_G \middle \vert G \in \mathbb L^* , \frac1{2} |G+k|^2 < E_{crit} \right \} + ``` + Note that thus a **different basis set** for each $k \in \mathbb K$ is employed. +""" + +# ╔═║ 9b7c42b4-9592-4ae3-8a86-41b6013885f3 + + +# ╔═║ e59d4d62-8ceb-43ad-ab86-159eac59500f +TableOfContents() + +# ╔═║ 7a952990-b524-43b6-a01b-cf6b6c6eaa14 +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 305) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +TikzPictures = "37f6aa50-8035-52d0-81c2-5a1d08754b2d" + +[compat] +HypertextLiteral = "~0.9.5" +LaTeXStrings = "~1.3.1" +PlutoTeachingTools = "~0.2.15" +PlutoUI = "~0.7.59" +TikzPictures = "~3.5.0" +""" + +# ╔═║ 00000000-0000-0000-0000-000000000002 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MATH 500 \ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 0000000..ca834da --- /dev/null +++ b/index.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/index.jl b/index.jl new file mode 100644 index 0000000..133298f --- /dev/null +++ b/index.jl @@ -0,0 +1,515 @@ +### A Pluto.jl notebook ### +# v0.19.43 + +using Markdown +using InteractiveUtils + +# ╔═║ a9d71171-dd24-472a-be5f-12619295c4bf +begin + using HypertextLiteral + using PlutoUI + using PlutoTeachingTools +end + +# ╔═║ 9261420d-3754-4820-a4f0-c5bda988157a +md""" +# MATH-500: Error control in scientific modelling +""" + +# ╔═║ 09579796-7e0e-4990-b727-b20f63242bcb +md""" +This repository contains the teaching material of the course +[MATH-500 Error control in scientific modelling](https://edu.epfl.ch/coursebook/en/error-control-in-scientific-modelling-MATH-500) at EPFL. +""" + +# ╔═║ c0da4104-9a5a-4553-8df0-fdadd0194c14 +md""" +## Course outline +""" + +# ╔═║ d448115a-6857-4082-893b-37549fe83b8b +md""" +If you spot an error feel free to make a pull request to the +[github repository](https://github.com/epfl-matmat/error-control-modelling) +generating this website. +""" + +# ╔═║ 854d367b-1609-4c00-9c62-9439874a03d6 +md""" +## EPFL resources + +If you are an EPFL student, you can find the following course resources: +- **Moodle link:** + +""" + +# ╔═║ ebf43572-95f2-401e-aa2e-d7748d3bf639 +md""" +## Summary +Errors are ubiquitous in computational science as neither models nor numerical +techniques are perfect. With respect to eigenvalue problems motivated from +materials science and atomistic modelling we discuss, +implement and apply numerical techniques for estimating simulation error. +""" + +# ╔═║ 7e571edb-6206-4138-863e-5a9d804e07ec +md""" +## Content +* Important eigenvalue problems in materials science +* Motivation for studying errors in eigenvalue problems +* Types of simulation error +* Residual-error relationships for eigenvalue problems +* Perturbation theory and parametrised eigenvalue problems +* Subtleties of infinite-dimensional eigenvalue problems +* Discretisation and discretisation error +* Plane-wave basis sets +* Errors due to uncertain parameters (if time permits) +* Non-linear eigenvalue problems (if time permits) + +Algorithm demonstrations and implementations will be based on +the [Julia programming language](https://julialang.org/) +and interactive [Pluto](https://plutojl.org/) notebooks. +""" + +# ╔═║ 0ccf77bd-0c5d-4c3c-82b5-b25c4adbc3ba +md""" +## Prerequisites +* Analysis +* Linear algebra +* Exposure to numerical linear algebra +* Some experience with numerical methods for solving differential equations + (such as finite-element methods, finite-difference approaches, plane-wave methods) +* Exposure to implementing numerical algorithms (e.g. using Python or Julia) + +Past participants from materials science found it further useful to take this course after they followed the lectures on [Statistical Mechanics](https://edu.epfl.ch/coursebook/fr/statistical-mechanics-MSE-421) or [Fundamentals of solid state materials](https://edu.epfl.ch/coursebook/en/fundamentals-of-solid-state-materials-MSE-423). + +This course delivers a mathematical viewpoint on materials modelling and it is +explicitly intended for an interdisciplinary student audience. To keep it +accessible, the key mathematical and physical concepts will both be revised as +we go along. However, the learning curve will be steep and an interest to learn +about the respective other discipline is required. + +The problem sheets and the +projects require a substantial amount of work and feature both theoretical +(proof-oriented) and applied (programming-based and simulation-based) components. +While there is some freedom for students to select their respective +focus, students are encouraged to team up across the disciplines +for the course work. +""" + +# ╔═║ b1c510c9-4130-4f5e-8d15-3bfaf0a61033 +md""" +## Literature +There is no single textbook corresponding to the content of the course. +Parts of the lectures have substantial overlap with the following resources, +where further information can be found. + +- Youssef Saad. *Numerical Methods for Large Eigenvalue Problems*, SIAM (2011). + A PDF is available free of charge on + [Youssef Saad's website](https://www-users.cse.umn.edu/~saad/eig_book_2ndEd.pdf). +- Nicholas J. Higham. *Accuracy and Stability of Numerical Algorithms*, SIAM (2002). +- Peter Arbenz. *Lecture notes on solving large scale eigenvalue problems*, ETHZ. + A PDF is available from + [Peter Arbenz' website](https://people.inf.ethz.ch/arbenz/ewp/Lnotes/lsevp.pdf). +- Mathieu Lewin. *ThΓ©orie spectrale et mΓ©canique quantique*, Springer (2022). +""" + +# ╔═║ 4e1fa8a0-ad16-417f-ba58-5cb0ccff45a3 +TableOfContents() + +# ╔═║ 8d38c66a-9582-4ef0-96f7-73e0b2ae2e2f +begin + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + toc = Markdown.parse(read("sidebar.md", String)) + Sidebar(toc, ypos) = @htl("""""") + Sidebar(toc, 315) +end + +# ╔═║ fefc5308-7caf-465a-afc0-b1a6f2e68ba2 +toc + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" + +[compat] 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H}} + +\def\hilbert{{\mathscr H}} +\def\graph{G} +\def\boundedoperators{\mathscr B} + +\def\bloch{\mathcal B} + +\def\indicator{{\mathbf 1}} + +\def\im{\operatorname{Im}} +\def\ker{\operatorname{Ker}} + +\definecolor{noteblue}{RGB}{123, 145, 178} +\definecolor{warnyellow}{RGB}{165, 159, 116} +\definecolor{prooftext}{RGB}{85, 85, 85} +``` diff --git a/nomenclature.html b/nomenclature.html new file mode 100644 index 0000000..622f4f2 --- /dev/null +++ b/nomenclature.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/nomenclature.jl b/nomenclature.jl new file mode 100644 index 0000000..1fa7d7d --- /dev/null +++ b/nomenclature.jl @@ -0,0 +1,516 @@ +### A Pluto.jl notebook ### +# v0.19.43 + +using Markdown +using InteractiveUtils + +# ╔═║ 6a1b2185-a0de-45f1-8d73-ab30f404c8c2 +begin + using HypertextLiteral + using PlutoUI + using PlutoTeachingTools + + RobustLocalResource("https://teaching.matmat.org/error-control/latex_macros.md", "latex_macros.md") + Markdown.parse(read("latex_macros.md", String)) +end + +# ╔═║ 48662bbd-d763-4e6b-957b-f4d6a783afc5 +md"# Nomenclature" + +# ╔═║ 506de2af-faea-42e4-8f14-166a9ad600ed +md""" +Latin | alphabet +--- | :--- +$A$ | Generic matrix +$\opA$ | Generic operator +$\bloch$ | Bloch-Floquet transform +$\mathbb B$ | Plane wave basis +$\mathscr B$ | Set of all bounded operators +$\contour$ | Contour in the complex plane +$D^\alpha$ | Weak derivative +$D(\opA)$ | Domain of $\opA$ +$\eigenspace _A(\lambda)$ | Eigenspace of $A$ associated with eigenvalue $\lambda$. +$G(\opA)$ | Graph of $\opA$ +$H$ | Sobolev space (see function spaces below) +$\opH$ | SchrΓΆdinger operator / hamiltonian ($- \laplacian / 2 + V$) +$\opH_k$ | Bloch fiber +$\hilbert$ | Hilbert space +$I$ | Identity matrix +$\mathbb K$ | $k$-grid or $k$-point mesh +$\mathbb L$ | Lattice +$\mathbb L^*$ | Reciprocal lattice +$q_A(u)$ | Quadratic form ($\langle u, Au \rangle$) +$Q(\opA)$ | Form domain of $\opA$ +$a_A(u,v)$ | Sesquilinear form ($\langle u, Av \rangle$) +$R_A(u)$ | Rayleigh quotient ($\langle u , Au \rangle / \langle u, u \rangle$) +$R_z(A)$ | Resolvent ($(A-z I)^{-1}$) +$\mathcal T_R$ | Translation operator by $R$ +""" + +# ╔═║ fc3cda3f-8ba8-4911-a8a4-ec46cf1b4dfb +md""" +Here is a non-comprehensive list of the notation used in the couse, with an emphasis on course-specific concepts, possible sources of confusion, and notation differing from typical physics / engineering notation. +""" + +# ╔═║ 7f43fd45-f6a5-449b-8401-627dcb047c80 +md""" +Greek | alphabet +--- | :--- +$\Delta$ | Laplace operator ($\text{div grad}$) +$\resolvent$ | Resovlent set +$\spectralradius$ | Spectral radius +$\sigma$ | Spectrum +$\Sigma$ | Bottom of the essential spectrum +""" + +# ╔═║ 1bb4d24e-f93c-4f9a-aaa3-1f799b5e2953 +md""" +Other | - +--- | :--- +$\bullet ^*$ | Adjoint (operators) +$\tilde \bullet$ | Approximation of $\bullet$ +$\overline{\bullet}$ | Complex conjugate (scalars), closure (sets) +$\dot \cup$ | Disjoint union +$\varnothing$ | Empty set +$\indicator_\Omega$ | Indicator function over set $\Omega$ +$\langle \bullet,\bullet \rangle$ | Inner product +$(\bullet,\bullet)$ | Open interval +$[\bullet, \bullet ]$ | Closed interval +$\leq$ | Vector subspace (sets), less or equal to (scalars) +$\to$ | Strong convergence +$\rightharpoonup$ | Weak convergence +""" + +# ╔═║ a1fc69f5-ce2e-4910-af4c-9db644a4dad5 +md""" +Function (and other) spaces + +Space | Definition | Hilbert space ? +--- | :--- | --- +$V_0$ | Space $V$ restricted to functions with compact support. +$C^0(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f \text{ is continuous} \}$ +$C^1(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f' \text{ is continuous} \}$ +$C^k(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f \text{ is infinitely differentiable} \}$ +$C^\infty(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f^{(k)} \text{ is continuous} \}$ +$L^2(\Omega)$ | $\{f : \Omega \to \mathbb C \mid \int_{\Omega} \vert f (x) \vert ^2 dx < \infty \}$ | βœ“ +$L^p(\Omega)$ | $\{f : \Omega \to \mathbb C \mid \int_{\Omega} \vert f (x) \vert ^p dx < \infty \}$ +$L^p_{loc}(\Omega)$ | $\left \{ f : \Omega \rightarrow \mathbb{C} \ \middle \vert \ f\rvert_K \in L^{p}(K) \quad \forall K \in \Omega, K \text { compact} \right \}$ +$L^2_{per} (\Omega)$ | $\{ f \in L^2_{loc} (\mathbb R^3) \vert f \text{ is } \mathbb L \text{ periodic and } \mathbb L \text{ has unit cell } \Omega \}$ | βœ“ +$L^2_{qp} (\Omega^*, L^2_{per}(\Omega))$ | $\{ \mathbb R^d \ni k \mapsto u_k \in L^2_{per}(\Omega) \vert \int_{\Omega^*} \| u_k \|^2_{L^2_{per}(Ξ©)} \ dk < ∞ \ \text{ and } u_{k+G} = u_k e^{-i G β‹… x} \}$ | βœ“ +$\mathscr L(V)$ | $\{ f : V \to V \mid f \text{ linear} \}$ +$\ell^2(\mathbb C)$ | $\{z : \mathbb N \to \mathbb C \mid \ \sum_{n = 0}^\infty \vert z_i \vert^2 < \infty \}$ | βœ“ +$\ell^p(\mathbb C)$ | $\{z : \mathbb N \to \mathbb C \mid \ \sum_{n = 0}^\infty \vert z_i \vert^p < \infty \}$ +$H^n(\Omega)$ | $\{ f \in L^2(\Omega) \mid D^\alpha f \in L^2(\Omega) \ \forall \alpha \text{ s.t. } \Vert \alpha \Vert _1 \leq n \}$ | βœ“ +$H^S_{per} (\Omega)$ | $\{ f \in L^2_{per} (\Omega) \vert \sum_{G \in \mathbb L^*} (1 + \vert G \vert^2)^S \vert \hat f_G \vert^2 < \infty \}$ | βœ“ + +Note that $\Omega$ here is used in most cases to denote the set on which the function is defined. +However, in the case of periodic function spaces ($L^2_{per}(\Omega), H^S_{per}(\Omega)$), it denotes the unit cell. +""" + +# ╔═║ 14976bf5-5f50-4920-b8d1-632c440be175 +md""" +Inner product of Hilbert spaces (for their definitions, see the table above). +To obtain the associated norm, recall $\| f \| = \sqrt{\langle f,f \rangle}$. + +Space | Inner Product $\langle f,g \rangle$ +--- | :--- | +$L^2(\Omega)$ | $\int_\Omega \overline{f(x)} g(x) \ dx$ +$\ell^2(\mathbb R)$ | $\sum_{i=0}^\infty \overline{f_i} g_i$ +$H^n(\Omega)$ | $\sum_{\Vert \alpha \Vert _1 \leq n} \langle D^\alpha f, D^\alpha g \rangle_{L^2}$ +$L^2_{per} (\Omega)$ | $\int_{\Omega} \overline{f(x)} g(x) dx$ +$L^2_{qp} (\Omega^*, H^1_{per}(\Omega))$ | $\frac1{\vert \Omega^*\vert} \int_{\Omega^*} \langle f_k, g_k \rangle_{L^2_{per}(\Omega)} dk$ +""" + +# ╔═║ a77b7363-f50f-45c9-a17b-d474b5b14baf +TableOfContents() + +# ╔═║ 7c87a6e1-dfe6-4bc1-8857-3b4a55aa6c0b +let + RobustLocalResource("https://teaching.matmat.org/error-control/sidebar.md", "sidebar.md") + Sidebar(toc, ypos) = @htl("""""") + + Sidebar(Markdown.parse(read("sidebar.md", String)), 75) +end + +# ╔═║ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = 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\ No newline at end of file diff --git a/sidebar.md b/sidebar.md new file mode 100644 index 0000000..59fb158 --- /dev/null +++ b/sidebar.md @@ -0,0 +1,14 @@ +**[Error control in scientific modelling](https://teaching.matmat.org/error-control/)** + +1. [Introduction](https://teaching.matmat.org/error-control/01_Introduction.html) +1. [The Dirichlet Laplacian in an Unbounded Domain](https://teaching.matmat.org/error-control/02_Laplace_error_sources.html) +1. [Matrix Eigenproblems](https://teaching.matmat.org/error-control/03_Matrix_eigenproblems.html) +1. [Bounds on Eigenvalues](https://teaching.matmat.org/error-control/04_Matrix_error_bounds.html) +1. [Errors due to floating-point arithmetic](https://teaching.matmat.org/error-control/05_Floating_point_arithmetic.html) +1. [Diagonalisation algorithms](https://teaching.matmat.org/error-control/06_Diagonalisation_algorithms.html) +1. [Matrix Perturbation Theory](https://teaching.matmat.org/error-control/07_Perturbation_theory.html) +1. [Hilbert Spaces](https://teaching.matmat.org/error-control/08_Hilbert_spaces.html) +1. [Operators](https://teaching.matmat.org/error-control/09_Operators.html) +1. [Self-adjoint operators](https://teaching.matmat.org/error-control/10_Spectra_self_adjoint.html) +1. [Periodic Problems](https://teaching.matmat.org/error-control/11_Periodic_problems.html) +1. [Nomenclature](https://teaching.matmat.org/error-control/nomenclature.html)