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<html>
<head>
<title>
SPARSE_GRID_CC - Sparse Grids Based on the Clenshaw Curtis Rule
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_CC <br> Sparse Grids Based on the Clenshaw Curtis Rule
</h1>
<hr>
<p>
<b>SPARSE_GRID_CC</b>
is a FORTRAN90 library which
can be used to compute the points and weights of a Smolyak sparse
grid, based on a 1-dimensional Clenshaw-Curtis quadrature rule,
to be used for efficient and accurate quadrature in multiple dimensions.
</p>
<p>
One way of looking at the construction of sparse grids is to assume
that we start out by constructing a (very dense) product grid.
We will assume for now that the <b>order</b>, that is, the number of
points used in each component grid, is the same for all dimensions.
Moreover, we will assume that the order is a power of 2 plus one,
so that we have a natural relationship between the order and
the logarithm base 2 of the order minus 1:
</p>
<p>
<pre><b>
order = 2<sup>level</sup> + 1
</b></pre>
except that for the special case of <b>level=0</b> we assign
<b>order=1</b>. (If we used our formula, then this case would
give us <b>order=2</b> instead.
</p>
<p>
Thus, if we allow <b>level</b> to grow, the <b>order</b> roughly
doubles, as follows:
<table border = "1">
<tr><th>Level</th><th>Order</th></tr>
<tr><td> 0</td><td> 1</th></tr>
<tr><td> 1</td><td> 3</th></tr>
<tr><td> 2</td><td> 5</th></tr>
<tr><td> 3</td><td> 9</th></tr>
<tr><td> 4</td><td> 17</th></tr>
<tr><td> 5</td><td> 33</th></tr>
<tr><td> 6</td><td> 65</th></tr>
<tr><td> 7</td><td> 129</th></tr>
<tr><td> 8</td><td> 257</th></tr>
<tr><td> 9</td><td> 513</th></tr>
<tr><td> 10</td><td> 1025</th></tr>
</table>
</p>
<p>
To keep things simple, let us begin by supposing we are selecting
points for a grid to be used in an interpolation or quadrature rule.
If you successively compute the locations of the points of each
level, you will probably see that the points of a level
are all included in the grid associated with the next level.
(This is not guaranteed for all rules; it's simply a property
of the way most such grids are defined!).
</p>
<p>
This <b>nesting</b> property is very useful. For one thing,
it means that when if we've computed a grid of one level, and now
proceed to the next, then all the information associated with
the current level (point location, the value of functions at those
points) is still useful for the next level, and will save us
some computation time as well. This also means that, when we
have reached a particular level, all the previous levels are
still available to us, with no extra storage. These considerations
make it possible, for instance, to do efficient and convenient
error estimation.
</p>
<p>
When we move to a problem whose geometry is two-dimensional or
more, we can still take the same approach. However, when working
in multidimensional geometry, it is usually not a good idea to
form a grid using the product of 1D grids, especially when we
are determining the order using the idea of levels. Especially
in this case, if we go to the next level in each dimension, the
total number of points would increase by a factor of roughly
2 to the spatial dimension. Just a few such steps in, say,
6 dimensions, and we would be far beyond our computational capacity.
</p>
<p>
Instead, in multidimensions, the idea is to construct a <i>sparse
grid</i>, which can be thought of in one of two ways:
<ul>
<li>
the sparse gird is a logical sum of low order product grids;
each product grid has a total level (sum of the levels of the
1d rules) that is less than or equal to <b>level_max</b>;
</li>
<li>
the sparse grid is a very sparse selection of points from the
very high order product grid formed by using rules of level
<b>level_max</b> in each dimension.
</li>
</ul>
</p>
<p><i>
(There is still a lot of explaining to do to get from the one-dimensional
levels to the N-dimensional levels and the selection of the low-level
product grids that sum up to the sparse grid...)
</i></p>
<p>
Once the grid indices of the sparse grid points have been selected,
there are a variety of schemes for distributing the points. We
consider closed quadrature rules, in which the endpoints of the
interval are included. The uniform scheme, known as the
Newton Cotes Closed rule, is easy to understand.
However, it has been observed that greater accuracy and stability
can be achieved by arranging the points in a nonuniform way that tends
to move points towards the boundary and away from the center.
A common scheme for doing this starts with the uniform points on
[0,1] and applies the cosine function to arrive at nonuniformly
spaced points in [-1,1]. This scheme is known as the
Clenshaw Curtis rule.
<p>
<p>
<ul>
<li>
Newton Cotes Closed: <br>
Points = [ 0, 1, 2, ..., N-1 ] / (N-1);<br>
Uniformly spaced points on [0,1], including endpoints.<br>
Point spacing is 1/(N-1).
</li>
<li>
Clenshaw Curtis:<br>
Theta = pi * [ 0, 1, 2, ..., N-1 ] / (N-1);<br>
Points = cos ( Theta );<br>
Nonuniformly spaced points on [-1,1], including endpoints.
</li>
</ul>
</p>
<p>
The library of routines presented here will only construct grids
based on the Clenshaw Curtis rule.
<p>
<h3 align = "center">
Licensing:
</h3>
<p>
The code described and made available on this web page is distributed
under the
<a href = "gnu_lgpl.txt">GNU LGPL</a> license.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPARSE_GRID_CC</b> is available in
<a href = "../../cpp_src/sparse_grid_cc/sparse_grid_cc.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_cc/sparse_grid_cc.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_cc/sparse_grid_cc.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
defines quadrature rules;
a number of examples of sparse grid quadrature rules are included.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules for
various intervals and weight functions.
</p>
<p>
<a href = "../../f_src/sgmga/sgmga.html">
SGMGA</a>,
a FORTRAN90 library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</p>
<p>
<a href = "../../c_src/smolpack/smolpack.html">
SMOLPACK</a>,
a C library which
implements Novak and Ritter's method for estimating the integral
of a function over a multidimensional hypercube using sparse grids.
</p>
<p>
<a href = "../../datasets/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Clenshaw Curtis rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_cc_dataset/sparse_grid_cc_dataset.html">
SPARSE_GRID_CC_DATASET</a>,
a FORTRAN90 program which
creates a sparse grid dataset based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Fejer Type 2 rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Legendre rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_gp/sparse_grid_gp.html">
SPARSE_GRID_GP</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Gauss Patterson rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../f_src/sparse_grid_laguerre/sparse_grid_laguerre.html">
SPARSE_GRID_LAGUERRE</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Laguerre rules.
</p>
<p>
<a href = "../../f_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a FORTRAN90 library which
constructs a sparse grid using different rules in each spatial dimension.
</p>
<p>
<a href = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes closed rule.
</p>
<p>
<a href = "../../datasets/sparse_grid_nco/sparse_grid_nco.html">
SPARSE_GRID_NCO</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes open rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_open/sparse_grid_open.html">
SPARSE_GRID_OPEN</a>,
a FORTRAN90 library which
creates sparse grids based on
open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).
</p>
<p>
<a href = "../../m_src/toms847/toms847.html">
TOMS847</a>,
a MATLAB program which
uses sparse grids to carry out multilinear hierarchical interpolation.
It is commonly known as SPINTERP, and is by Andreas Klimke.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Volker Barthelmann, Erich Novak, Klaus Ritter,<br>
High Dimensional Polynomial Interpolation on Sparse Grids,<br>
Advances in Computational Mathematics,<br>
Volume 12, Number 4, 2000, pages 273-288.
</li>
<li>
Thomas Gerstner, Michael Griebel,<br>
Numerical Integration Using Sparse Grids,<br>
Numerical Algorithms,<br>
Volume 18, Number 3-4, 1998, pages 209-232.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 46, Number 5, 2008, pages 2309-2345.
</li>
<li>
Sergey Smolyak,<br>
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,<br>
Doklady Akademii Nauk SSSR,<br>
Volume 4, 1963, pages 240-243.
</li>
<li>
Dennis Stanton, Dennis White,<br>
Constructive Combinatorics,<br>
Springer, 1986,<br>
ISBN: 0387963472,<br>
LC: QA164.S79.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_cc.f90">sparse_grid_cc.f90</a>, the source code.
</li>
<li>
<a href = "sparse_grid_cc.sh">sparse_grid_cc.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_cc_prb.f90">sparse_grid_cc_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "sparse_grid_cc_prb.sh">sparse_grid_cc_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_cc_prb_output.txt">sparse_grid_cc_prb_output.txt</a>,
the output from a run of the sample program.
</li>
<li>
<a href = "cc_d4_level2_r.txt">cc_d4_level2_r.txt</a>,
the "R" file for a sparse grid quadrature rule for spatial dimension 4
and level 2.
</li>
<li>
<a href = "cc_d4_level2_w.txt">cc_d4_level2_w.txt</a>,
the "W" file for a sparse grid quadrature rule for spatial dimension 4
and level 2.
</li>
<li>
<a href = "cc_d4_level2_x.txt">cc_d4_level2_x.txt</a>,
the "X" file for a sparse grid quadrature rule for spatial dimension 4
and level 2.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ABSCISSA_LEVEL_CLOSED_ND:</b> first level at which given abscissa is generated.
</li>
<li>
<b>CC_ABSCISSA</b> returns the I-th abscissa for the Clenshaw Curtis rule.
</li>
<li>
<b>CC_WEIGHTS</b> computes Clenshaw Curtis weights.
</li>
<li>
<b>CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>INDEX_TO_LEVEL_CLOSED</b> determines the level of a point given its index.
</li>
<li>
<b>LEVEL_TO_ORDER_CLOSED</b> converts a level to an order for closed rules.
</li>
<li>
<b>LEVELS_CLOSED_INDEX</b> computes closed grids with 0 <= LEVEL <= LEVEL_MAX.
</li>
<li>
<b>MONOMIAL_INT01</b> returns the integral of a monomial over the [0,1] hypercube.
</li>
<li>
<b>MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>MULTIGRID_INDEX0</b> returns an indexed multidimensional grid.
</li>
<li>
<b>MULTIGRID_SCALE_CLOSED</b> renumbers a grid as a subgrid on a higher level.
</li>
<li>
<b>PRODUCT_WEIGHTS_CC:</b> Clenshaw Curtis product rule weights.
</li>
<li>
<b>R8_HUGE</b> returns a very large R8.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>S_BLANK_DELETE</b> removes blanks from a string, left justifying the remainder.
</li>
<li>
<b>SPARSE_GRID_CC</b> computes a sparse grid of Clenshaw Curtis points.
</li>
<li>
<b>SPARSE_GRID_CC_INDEX</b> indexes the points forming a sparse grid.
</li>
<li>
<b>SPARSE_GRID_CFN_SIZE</b> sizes a sparse grid using Closed Fully Nested rules.
</li>
<li>
<b>SPARSE_GRID_CC_SIZE_OLD</b> sizes a sparse grid based on nested closed rules.
</li>
<li>
<b>SPARSE_GRID_CC_WEIGHTS</b> gathers the weights.
</li>
<li>
<b>SPARSE_GRID_CCS_SIZE</b> sizes a sparse grid using Clenshaw Curtis Slow rules.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VEC_COLEX_NEXT2</b> generates vectors in colex order.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 22 December 2009.
</i>
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