SPARSE_GRID_CC_DATASET
Create ClenshawCurtis Sparse Grid Dataset
SPARSE_GRID_CC_DATASET
is a C++ program which
creates a ClenshawCurtis sparse grid dataset.
Usage:
sparse_grid_cc_dataset dim_num level_max
where

dim_num is the spatial dimension, typically between 2 and 10;

level_max is the sparse grid level, typically between 0 and 6,
which controls the number of points
in the grid. The 1D rules used will have order 2^(level_max)+1.
Each sparse grid is stored using the "quadrature rule" format,
that is, as three files:

an "R" or "region" file, which lists two points that bound the region;

a "W" or "weight" file, which lists the weight for each abscissa;

an "X" or "abscissa" file, which lists the abscissas of the rule.
The abscissas are ordered to respect the natural nesting of the
sparse grids by level. That is, the file of level 3 points begins by
listing the points in the level 2 grid.
Licensing:
The code described and made available on this web page is distributed
under the
GNU LGPL license.
Languages:
SPARSE_GRID_CC_DATASET is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
CC_DISPLAY,
a MATLAB library which
can compute and display Clenshaw Curtis grids in two dimensions,
as well as sparse grids formed from sums of Clenshaw Curtis grids.
QUADRATURE_RULES,
a dataset directory of files which
defines quadrature rules;
a number of examples of sparse grid quadrature rules are included.
SGMGA,
a C++ library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
SMOLPACK,
a C library which
implements Novak and Ritter's method for estimating the integral
of a function over a multidimensional hypercube using sparse grids.
SPARSE_GRID_CC,
a C++ library which
creates a ClenshawCurtis sparse grid.
SPARSE_GRID_CC,
a dataset directory which
contains examples of ClenshawCurtis sparse grids.
SPARSE_GRID_F2,
a dataset directory which
conains the abscissas of sparse
grids based on a Fejer Type 2 rule.
SPARSE_GRID_GL_DATASET,
a C++ program which
creates a GaussLegendre sparse grid and writes the data to three files.
SPARSE_GRID_GP,
a dataset directory which
contains the abscissas of sparse
grids based on a Gauss Patterson rule.
SPARSE_GRID_HERMITE_DATASET,
a C++ program which
creates a GaussHermite sparse grid and write the data to three files.
SPARSE_GRID_LAGUERRE_DATASET,
a C++ program which
creates a GaussLaguerre sparse grid and write the data to three files.
SPARSE_GRID_MIXED_DATASET,
a C++ program which
creates a sparse grid dataset based on a mixture of 1D rules.
SPARSE_GRID_NCC,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes closed rule.
SPARSE_GRID_NCO,
a dataset directory which
contains the abscissas of sparse
grids based on a Newton Cotes open rule.
SPARSE_GRID_OPEN_DATASET,
a C++ program which
creates a sparse grid dataset based on
open rules (Fejer 2, GaussPatterson, NewtonCotesOpen).
TOMS847,
a MATLAB program which
uses sparse grids to carry out multilinear hierarchical interpolation.
It is commonly known as SPINTERP, and is by Andreas Klimke.
Reference:

Volker Barthelmann, Erich Novak, Klaus Ritter,
High Dimensional Polynomial Interpolation on Sparse Grids,
Advances in Computational Mathematics,
Volume 12, Number 4, 2000, pages 273288.

Thomas Gerstner, Michael Griebel,
Numerical Integration Using Sparse Grids,
Numerical Algorithms,
Volume 18, Number 34, 1998, pages 209232.

Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0125192606,
LC: QA164.N54.

Fabio Nobile, Raul Tempone, Clayton Webster,
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 46, Number 5, 2008, pages 23092345.

Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,
Doklady Akademii Nauk SSSR,
Volume 4, 1963, pages 240243.

Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.
Source Code:
Examples and Tests:
Here are the files created by the command
sparse_grid_cc_dataset 2 3
List of Routines:

MAIN is the main program for SPARSE_GRID_CC_DATASET.

ABSCISSA_LEVEL_CLOSED_ND: first level at which an abscissa is generated.

CC_ABSCISSA returns the Ith abscissa of the Clenshaw Curtis rule.

CC_WEIGHTS computes Clenshaw Curtis weights.

CHOOSE computes the binomial coefficient C(N,K).

COMP_NEXT computes the compositions of the integer N into K parts.

I4_MAX returns the maximum of two I4's.

I4_MIN returns the smaller of two I4's.

I4_MODP returns the nonnegative remainder of integer division.

I4_POWER returns the value of I^J.

I4_TO_STRING converts an I4 to a C++ string.

I4VEC_PRODUCT multiplies the entries of an integer vector.

INDEX_TO_LEVEL_CLOSED determines the level of a point given its index.

LEVEL_TO_ORDER_CLOSED converts a level to an order for closed rules.

MULTIGRID_INDEX0 returns an indexed multidimensional grid.

MULTIGRID_SCALE_CLOSED renumbers a grid as a subgrid on a higher level.

PRODUCT_WEIGHTS_CC computes weights for a Clenshaw Curtis product rule.

R8_EPSILON returns the R8 roundoff unit.

R8_HUGE returns a "huge" R8.

R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.

R8MAT_WRITE writes an R8MAT file with no header.

R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.

R8VEC_PRINT_SOME prints "some" of an R8VEC.

S_LEN_TRIM returns the length of a string to the last nonblank.

SPARSE_GRID_CC computes a sparse grid of Clenshaw Curtis points.

SPARSE_GRID_CC_INDEX indexes the points forming a sparse grid.

SPARSE_GRID_CFN_SIZE sizes a sparse grid of Closed Fully Nested points.

SPARSE_GRID_CC_WEIGHTS gathers the weights.

TIMESTAMP prints the current YMDHMS date as a time stamp.

VEC_COLEX_NEXT2 generates vectors in colex order.
You can go up one level to
the C++ source codes.
Last revised on 12 January 2010