Update week9.do.txt

doc/src/week9/week9.do.txt

@@ -7,9 +7,9 @@ DATE: March 11-15
 ===== Overview of week 11, March 11-15 =====
 !bblock  Topics
 o Reminder from last week about statistical observables, the central limit theorem and bootstrapping, see notes from last week
-o Resampling TechniquesL Blocking 
-o Discussion of onebody densities
-o Start discussion on optimization and parallelization
+o Resampling Techniques, emphasis on  Blocking 
+o Discussion of onebody densities (whiteboard notes)
+o Start discussion on optimization and parallelization for Python and C++
 #* "Video of lecture TBA":"https://youtu.be/"
 #* "Handwritten notes":"https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/HandWrittenNotes/2024/NotesMarch22.pdf"
 !eblock
@@ -167,7 +167,7 @@ in terms of a function
 f_d=\frac{2}{mn}\sum_{i=1}^{m} \sum_{k=1}^{n-d}\tilde{x}_{ik}\tilde{x}_{i(k+d)}.
 \]
 !et
-We note that for $d=$ we have
+We note that for $d=0$ we have
 !bt
 \[
 f_0=\frac{2}{mn}\sum_{i=1}^{m} \sum_{k=1}^{n}\tilde{x}_{ik}\tilde{x}_{i(k)}=\sigma^2!
@@ -187,304 +187,6 @@ We introduce then a correlation function $\kappa_d=f_d/\sigma^2$. Note that $\ka
 The code here shows the evolution of $\kappa_d$ as a function of $d$ for a series of random numbers. We see that the function $\kappa_d$ approaches $0$ as $d\rightarrow \infty$.
 
 
-!split
-===== Statistics, wrapping up from last week =====
-!bblock
-Let us analyze the problem by splitting up the correlation term into
-partial sums of the form:
-!bt
-\[
-f_d = \frac{1}{n-d}\sum_{k=1}^{n-d}(x_k - \bar x_n)(x_{k+d} - \bar x_n)
-\]
-!et
-The correlation term of the error can now be rewritten in terms of
-$f_d$
-!bt
-\[
-\frac{2}{n}\sum_{k<l} (x_k - \bar x_n)(x_l - \bar x_n) =
-2\sum_{d=1}^{n-1} f_d
-\]
-!et
-The value of $f_d$ reflects the correlation between measurements
-separated by the distance $d$ in the sample samples.  Notice that for
-$d=0$, $f$ is just the sample variance, $\mathrm{var}(x)$. If we divide $f_d$
-by $\mathrm{var}(x)$, we arrive at the so called *autocorrelation function*
-!bt
-\[
-\kappa_d = \frac{f_d}{\mathrm{var}(x)}
-\]
-!et
-which gives us a useful measure of pairwise correlations
-starting always at $1$ for $d=0$.
-!eblock
-
-
-!split
-===== Statistics, final expression =====
-!bblock
-The sample error can now be
-written in terms of the autocorrelation function:
-
-!bt
-\begin{align}
-\mathrm{err}_X^2 &=
-\frac{1}{n}\mathrm{var}(x)+\frac{2}{n}\cdot\mathrm{var}(x)\sum_{d=1}^{n-1}
-\frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=&
-\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\
-&=\frac{\tau}{n}\cdot\mathrm{var}(x)
-\end{align}
-
-!et
-and we see that $\mathrm{err}_X$ can be expressed in terms the
-uncorrelated sample variance times a correction factor $\tau$ which
-accounts for the correlation between measurements. We call this
-correction factor the *autocorrelation time*:
-!bt
-\begin{equation}
-\tau = 1+2\sum_{d=1}^{n-1}\kappa_d
-label{eq:autocorrelation_time}
-\end{equation}
-!et
-!eblock
-
-!split
-===== Statistics, effective number of correlations =====
-!bblock
-For a correlation free experiment, $\tau$
-equals 1.
-
-We can interpret a sequential
-correlation as an effective reduction of the number of measurements by
-a factor $\tau$. The effective number of measurements becomes:
-!bt
-\[
-n_\mathrm{eff} = \frac{n}{\tau}
-\]
-!et
-To neglect the autocorrelation time $\tau$ will always cause our
-simple uncorrelated estimate of $\mathrm{err}_X^2\approx \mathrm{var}(x)/n$ to
-be less than the true sample error. The estimate of the error will be
-too *good*. On the other hand, the calculation of the full
-autocorrelation time poses an efficiency problem if the set of
-measurements is very large.
-!eblock
-
-
-
-
-
-
-!split
-===== Can we understand this? Time Auto-correlation Function =====
-!bblock
-
-The so-called time-displacement autocorrelation $\phi(t)$ for a quantity $\mathbf{M}$ is given by
-!bt
-\[
-\phi(t) = \int dt' \left[\mathbf{M}(t')-\langle \mathbf{M} \rangle\right]\left[\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle\right],
-\]
-!et
-which can be rewritten as 
-!bt
-\[
-\phi(t) = \int dt' \left[\mathbf{M}(t')\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle^2\right],
-\]
-!et
-where $\langle \mathbf{M} \rangle$ is the average value and
-$\mathbf{M}(t)$ its instantaneous value. We can discretize this function as follows, where we used our
-set of computed values $\mathbf{M}(t)$ for a set of discretized times (our Monte Carlo cycles corresponding to moving all electrons?)
-!bt
-\[
-\phi(t)  = \frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\mathbf{M}(t'+t)
--\frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\times
-\frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t'+t).
-label{eq:phitf}
-\]
-!et
-!eblock
-
-
-!split
-===== Time Auto-correlation Function =====
-!bblock
-
-One should be careful with times close to $t_{\mathrm{max}}$, the upper limit of the sums 
-becomes small and we end up integrating over a rather small time interval. This means that the statistical
-error in $\phi(t)$ due to the random nature of the fluctuations in $\mathbf{M}(t)$ can become large.
-
-One should therefore choose $t \ll t_{\mathrm{max}}$.
-
-Note that the variable $\mathbf{M}$ can be any expectation values of interest.
-
-
-
-The time-correlation function gives a measure of the correlation between the various values of the variable 
-at a time $t'$ and a time $t'+t$. If we multiply the values of $\mathbf{M}$ at these two different times,
-we will get a positive contribution if they are fluctuating in the same direction, or a negative value
-if they fluctuate in the opposite direction. If we then integrate over time, or use the discretized version of, the time correlation function $\phi(t)$ should take a non-zero value if the fluctuations are 
-correlated, else it should gradually go to zero. For times a long way apart 
-the different values of $\mathbf{M}$  are most likely 
-uncorrelated and $\phi(t)$ should be zero.
-!eblock
-
-
-
-
-!split
-===== Time Auto-correlation Function =====
-!bblock
-We can derive the correlation time by observing that our Metropolis algorithm is based on a random
-walk in the space of all  possible spin configurations. 
-Our probability 
-distribution function $\mathbf{\hat{w}}(t)$ after a given number of time steps $t$ could be written as
-!bt
-\[
-   \mathbf{\hat{w}}(t) = \mathbf{\hat{W}^t\hat{w}}(0),
-\]
-!et
-with $\mathbf{\hat{w}}(0)$ the distribution at $t=0$ and $\mathbf{\hat{W}}$ representing the 
-transition probability matrix. 
-We can always expand $\mathbf{\hat{w}}(0)$ in terms of the right eigenvectors of 
-$\mathbf{\hat{v}}$ of $\mathbf{\hat{W}}$ as 
-!bt
-\[
-    \mathbf{\hat{w}}(0)  = \sum_i\alpha_i\mathbf{\hat{v}}_i,
-\]
-!et
-resulting in 
-!bt
-\[
-   \mathbf{\hat{w}}(t) = \mathbf{\hat{W}}^t\mathbf{\hat{w}}(0)=\mathbf{\hat{W}}^t\sum_i\alpha_i\mathbf{\hat{v}}_i=
-\sum_i\lambda_i^t\alpha_i\mathbf{\hat{v}}_i,
-\]
-!et
-with $\lambda_i$ the $i^{\mathrm{th}}$ eigenvalue corresponding to  
-the eigenvector $\mathbf{\hat{v}}_i$. 
-!eblock
-
-
-
-
-!split
-===== Time Auto-correlation Function =====
-!bblock
-If we assume that $\lambda_0$ is the largest eigenvector we see that in the limit $t\rightarrow \infty$,
-$\mathbf{\hat{w}}(t)$ becomes proportional to the corresponding eigenvector 
-$\mathbf{\hat{v}}_0$. This is our steady state or final distribution. 
-
-We can relate this property to an observable like the mean energy.
-With the probabilty $\mathbf{\hat{w}}(t)$ (which in our case is the squared trial wave function) we
-can write the expectation values as 
-!bt
-\[
- \langle \mathbf{M}(t) \rangle  = \sum_{\mu} \mathbf{\hat{w}}(t)_{\mu}\mathbf{M}_{\mu},
-\] 
-!et 
-or as the scalar of a  vector product
-!bt
- \[
- \langle \mathbf{M}(t) \rangle  = \mathbf{\hat{w}}(t)\mathbf{m},
-\] 
-!et 
-with $\mathbf{m}$ being the vector whose elements are the values of $\mathbf{M}_{\mu}$ in its 
-various microstates $\mu$.
-!eblock
-
-
-!split
-===== Time Auto-correlation Function =====
-
-!bblock
-
-We rewrite this relation  as
-!bt
- \[
- \langle \mathbf{M}(t) \rangle  = \mathbf{\hat{w}}(t)\mathbf{m}=\sum_i\lambda_i^t\alpha_i\mathbf{\hat{v}}_i\mathbf{m}_i.
-\] 
-!et 
-If we define $m_i=\mathbf{\hat{v}}_i\mathbf{m}_i$ as the expectation value of
-$\mathbf{M}$ in the $i^{\mathrm{th}}$ eigenstate we can rewrite the last equation as
-!bt
- \[
- \langle \mathbf{M}(t) \rangle  = \sum_i\lambda_i^t\alpha_im_i.
-\] 
-!et
-Since we have that in the limit $t\rightarrow \infty$ the mean value is dominated by the 
-the largest eigenvalue $\lambda_0$, we can rewrite the last equation as
-!bt
- \[
- \langle \mathbf{M}(t) \rangle  = \langle \mathbf{M}(\infty) \rangle+\sum_{i\ne 0}\lambda_i^t\alpha_im_i.
-\] 
-!et
-We define the quantity
-!bt
-\[
-   \tau_i=-\frac{1}{log\lambda_i},
-\]
-!et
-and rewrite the last expectation value as
-!bt
- \[
- \langle \mathbf{M}(t) \rangle  = \langle \mathbf{M}(\infty) \rangle+\sum_{i\ne 0}\alpha_im_ie^{-t/\tau_i}.
-label{eq:finalmeanm}
-\] 
-!et
-!eblock
-
-
-!split
-===== Time Auto-correlation Function =====
-!bblock
-
-The quantities $\tau_i$ are the correlation times for the system. They control also the auto-correlation function 
-discussed above.  The longest correlation time is obviously given by the second largest
-eigenvalue $\tau_1$, which normally defines the correlation time discussed above. For large times, this is the 
-only correlation time that survives. If higher eigenvalues of the transition matrix are well separated from 
-$\lambda_1$ and we simulate long enough,  $\tau_1$ may well define the correlation time. 
-In other cases we may not be able to extract a reliable result for $\tau_1$. 
-Coming back to the time correlation function $\phi(t)$ we can present a more general definition in terms
-of the mean magnetizations $ \langle \mathbf{M}(t) \rangle$. Recalling that the mean value is equal 
-to $ \langle \mathbf{M}(\infty) \rangle$ we arrive at the expectation values
-!bt
-\[
-\phi(t) =\langle \mathbf{M}(0)-\mathbf{M}(\infty)\rangle \langle \mathbf{M}(t)-\mathbf{M}(\infty)\rangle,
-\]
-!et
-resulting in
-!bt
-\[
-\phi(t) =\sum_{i,j\ne 0}m_i\alpha_im_j\alpha_je^{-t/\tau_i},
-\]
-!et
-which is appropriate for all times.
-!eblock
-
-
-
-!split
-===== Correlation Time =====
-!bblock
-
-If the correlation function decays exponentially
-!bt
-\[ \phi (t) \sim \exp{(-t/\tau)}\]
-!et
-then the exponential correlation time can be computed as the average
-!bt
-\[   \tau_{\mathrm{exp}}  =  -\langle  \frac{t}{log|\frac{\phi(t)}{\phi(0)}|} \rangle. \]
-!et
-If the decay is exponential, then
-!bt
-\[  \int_0^{\infty} dt \phi(t)  = \int_0^{\infty} dt \phi(0)\exp{(-t/\tau)}  = \tau \phi(0),\] 
-!et
-which  suggests another measure of correlation
-!bt
-\[   \tau_{\mathrm{int}} = \sum_k \frac{\phi(k)}{\phi(0)}, \]
-!et
-called the integrated correlation time.
-!eblock
-
-
 
 
 !split