SPARSE_GRID_CC_DATASET
Create Clenshaw-Curtis Sparse Grid Dataset

SPARSE_GRID_CC_DATASET is a MATlAB program which creates a Clenshaw-Curtis sparse grid dataset.

Usage:

sparse_grid_cc_dataset ( dim_num, level_max )

• 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

SPARSE_GRID_CC_DATASET is available in 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.

GRID_DISPLAY, a MATLAB library which can display a 2D or 3D grid or sparse grid.

SPARSE_GRID_CC, a MATLAB library which creates a Clenshaw-Curtis sparse grid.

SPARSE_GRID_CC, a dataset directory which contains examples of Clenshaw-Curtis sparse grids.

SPARSE_GRID_GL_DATASET, a MATLAB program which creates a Gauss-Legendre sparse grid and write the data to three files.

SPARSE_GRID_HERMITE_DATASET, a MATLAB program which creates a Gauss-Hermite sparse grid and write the data to three files.

SPARSE_GRID_LAGUERRE_DATASET, a MATLAB program which creates a Gauss-Laguerre sparse grid and write the data to three files.

SPARSE_GRID_MIXED_DATASET, a MATLAB program which creates a sparse grid dataset based on a mixture of 1D rules.

SPARSE_GRID_OPEN_DATASET, a MATLAB program which creates a sparse grid dataset based on open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).

Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
   High Dimensional Polynomial Interpolation on Sparse Grids,
   Advances in Computational Mathematics,
   Volume 12, Number 4, 2000, pages 273-288.
2. Thomas Gerstner, Michael Griebel,
   Numerical Integration Using Sparse Grids,
   Numerical Algorithms,
   Volume 18, Number 3-4, 1998, pages 209-232.
3. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
4. 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 2309-2345.
5. Sergey Smolyak,
   Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• sparse_grid_cc_dataset.m the program.

Examples and Tests:

Here are the three quadrature files created by the command

sparse_grid_cc_dataset 2 3

• sparse_grid_cc_dataset_output.txt, the output file.
• cc_d2_level3_r.txt, the "R" or "region" file.
• cc_d2_level3_w.txt, the "W" or "weight" file.
• cc_d2_level3_x.txt, the "X" or "abscissa" file.

You can go up one level to the MATLAB source codes.