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<html>
<head>
<title>
SPARSE_GRID_COMPOSITE - Sparse Grids Based on 1D Composite Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_COMPOSITE <br> Sparse Grids Based on 1D Composite Rules
</h1>
<hr>
<p>
<b>SPARSE_GRID_COMPOSITE</b>
is a MATLAB library which
can be used to compute the points and weights of a Smolyak sparse
grid, based on a 1-dimensional composite quadrature rule.
</p>
<p>
Currently, the library only allows composite rules of the 1 point
midpoint formula. A 1D composite rule of the midpoint rule is suitable
for integration of functions with low or variable smoothness. A sparse
grid formed from such rules can better handle multidimensional integrands
with limited smoothness.
</p>
<p>
Of course, for both the 1D composite rule
and the sparse grid rule, the repeated use of the midpoint rule limits
the order of accuracy of the quadrature; therefore, if the integrand is
known to be of higher smoothness, other approaches will produce an
accurate answer more quickly.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPARSE_GRID_COMPOSITE</b> is available in
<a href = "../../m_src/sparse_grid_composite/sparse_grid_composite.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/grid_display/grid_display.html">
GRID_DISPLAY</a>,
a MATLAB library which
can display a 2D or 3D grid or sparse grid.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory of files which
define quadrature rules;
a number of examples of sparse grid quadrature rules are included.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
defines quadrature rules for
various intervals and weight functions.
</p>
<p>
<a href = "../../m_src/sgmga/sgmga.html">
SGMGA</a>,
a MATLAB 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 = "../../m_src/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a MATLAB library which
creates sparse grids based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_composite/sparse_grid_composite.html">
SPARSE_GRID_COMPOSITE</a>,
a dataset directory which
contains examples of sparse grids based on 1D composite rules (currently only of order 1).
</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 = "../../m_src/sparse_grid_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a MATLAB library which
computes sparse
grids based on a Gauss-Legendre rule.
</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 = "../../m_src/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a MATLAB library which
creates sparse grids based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../m_src/sparse_grid_hw/sparse_grid_hw.html">
SPARSE_GRID_HW</a>,
a MATLAB library which
creates sparse grids based on Gauss-Legendre, Gauss-Hermite,
Gauss-Patterson, or a nested variation of Gauss-Hermite rules,
by Florian Heiss and Viktor Winschel.
</p>
<p>
<a href = "../../m_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a MATLAB 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 = "../../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 = "abscissa_level_closed_nd.m">
abscissa_level_closed_nd.m</a>
first level at which an abscissa is generated.
</li>
<li>
<a href = "comp_next.m">
comp_next.m</a>
computes the compositions of the integer N into K parts.
</li>
<li>
<a href = "composite_abscissa.m">
composite_abscissa.m</a>
returns the I-th abscissa of the composite rule.
</li>
<li>
<a href = "composite_weights.m">
composite_weights.m</a>
computes composite rule weights.
</li>
<li>
<a href = "i4_choose.m">
i4_choose.m</a>
computes the binomial coefficient C(N,K).
</li>
<li>
<a href = "i4_modp.m">
i4_modp.m</a>
returns the nonnegative remainder of integer division.
</li>
<li>
<a href = "index_to_level_closed.m">
index_to_level_closed.m</a>
determines the level of a point given its index.
</li>
<li>
<a href = "level_to_order_closed.m">
level_to_order_closed.m</a>
converts a level to an order for closed rules.
</li>
<li>
<a href = "monomial_int01.m">
monomial_int01.m</a>
returns the integral of a monomial over the [0,1] hypercube.
</li>
<li>
<a href = "monomial_quadrature.m">
monomial_quadrature.m</a>
applies a quadrature rule to a monomial.
</li>
<li>
<a href = "monomial_value.m">
monomial_value.m</a>
evaluates a monomial.
</li>
<li>
<a href = "multigrid_index0.m">
multigrid_index0.m</a>
returns an indexed multidimensional grid.
</li>
<li>
<a href = "multigrid_scale_closed.m">
multigrid_scale_closed.m</a>
renumbers a grid as a subgrid on a higher level.
</li>
<li>
<a href = "product_weights_composite.m">
product_weights_composite.m</a>
computes weights for a composite product rule.
</li>
<li>
<a href = "r8mat_write.m">
r8mat_write.m</a>
writes a table file.
</li>
<li>
<a href = "r8vec_direct_product2.m">
r8vec_direct_product2.m</a>
direct product of double precision real vectors.
</li>
<li>
<a href = "sparse_grid_composite.m">
sparse_grid_composite.m</a>
computes a sparse grid of composite rule points.
</li>
<li>
<a href = "sparse_grid_composite_index.m">
sparse_grid_composite_index.m</a>
indexes the points forming a sparse grid.
</li>
<li>
<a href = "sparse_grid_composite_size.m">
sparse_grid_composite_size.m</a>
sizes a sparse grid of composite rule points.
</li>
<li>
<a href = "sparse_grid_composite_weights.m">
sparse_grid_composite_weights.m</a>
gathers the weights.
</li>
<li>
<a href = "timestamp.m">
timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "vec_colex_next2.m">
vec_colex_next2.m</a>
generates vectors in colex order.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_composite_test.m">sparse_grid_composite_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "sparse_grid_composite_test01.m">sparse_grid_composite_test01.m</a>,
tests SPARSE_GRID_COMPOSITE_SIZE.
</li>
<li>
<a href = "sparse_grid_composite_test02.m">sparse_grid_composite_test02.m</a>,
tests SPARSE_GRID_COMPOSITE_INDEX.
</li>
<li>
<a href = "sparse_grid_composite_test03.m">sparse_grid_composite_test03.m</a>,
tests SPARSE_GRID_COMPOSITE.
</li>
<li>
<a href = "sparse_grid_composite_test04.m">sparse_grid_composite_test04.m</a>,
computes the weights and sums them.
</li>
<li>
<a href = "sparse_grid_composite_test05.m">sparse_grid_composite_test05.m</a>,
tests a COMPOSITE sparse grid rule for exactness on monomials.
</li>
<li>
<a href = "sparse_grid_composite_test06.m">sparse_grid_composite_test06.m</a>,
writes a COMPOSITE sparse grid rule to three files.
</li>
<li>
<a href = "composite_d2_level3_r.txt">composite_d2_level3_r.txt</a>,
the "R" data for a sparse COMPOSITE quadrature rule.
</li>
<li>
<a href = "composite_d2_level3_w.txt">composite_d2_level3_w.txt</a>,
the "W" data for a sparse COMPOSITE quadrature rule.
</li>
<li>
<a href = "composite_d2_level3_x.txt">composite_d2_level3_x.txt</a>,
the "X" data for a sparse COMPOSITE quadrature rule.
</li>
<li>
<a href = "sparse_grid_composite_test_output.txt">
sparse_grid_composite_test_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 16 September 2011.
</i>
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