Sparse Grid Files
Gauss Patterson

These are some examples of sparse grids, using the idea of a level to control the number of points, and assigning point locations using the Gauss Patterson rule.

The file has a simple structure. There are a few initial comment lines, which begin with the "#" character. These can be removed if they are inconvenient. There then follow the coordinates of the points, with each point on a single line, and the coordinates of a point separated by spaces.

The points are ordered to respect the natural nesting of the sparse grids. That is, the file of level 3 points begins by listing the points in the level 2 grid.

To make full use of such a sparse grid, it is also necessary to compute the associated interpolation weights.

Example:

Here is the text of a file storing a Gauss Patterson rule of level 1.

      0.000000      0.000000
     -0.774596      0.000000
      0.774596      0.000000
      0.000000     -0.774596
      0.000000      0.774596
      

Related Data and Programs:

• SMOLPACK is a C library of routines which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.
• SPARSE_GRID_DISPLAY is a library of MATLAB routines which can display a 2D or 3D sparse grid.
• SPINTERP is a MATLAB library of routines by Andreas Klimke which use a sparse grid to perform multilinear hierarchical interpolation.
• TABLE is the file format used to store this data;

• QUADRATURE_RULES is a dataset directory of files that define quadrature rules; a number of examples of sparse grid quadrature rules are included.
• SPARSE_GRID_CC some sample sparse grids from the Clenshaw Curtis rule.
• SPARSE_GRID_F1 some sample sparse grids from the Fejer Type 1 rule.
• SPARSE_GRID_F2 some sample sparse grids from the Fejer Type 2 rule.
• SPARSE_GRID_NCC some sample sparse grids from the Newton Cotes Closed rule.
• SPARSE_GRID_NCO some sample sparse grids from the Newton Cotes Open rule.
• SPARSE_GRID_NCOH some sample sparse grids from the Newton Cotes Open Half rule.

Sample Files:

Some grids in 2D:

• gp_d2_level0.txt, Gauss Patterson rule in 2 dimensions, level 0, in [-1,1], 1 point.
• gp_d2_level1.txt, Gauss Patterson rule in 2 dimensions, level 1, in [-1,1], 5 points.
• gp_d2_level2.txt, Gauss Patterson rule in 2 dimensions, level 2, in [-1,1], 17 points.
• gp_d2_level3.txt, Gauss Patterson rule in 2 dimensions, level 3, in [-1,1], 49 points.
• gp_d2_level4.txt, Gauss Patterson rule in 2 dimensions, level 4, in [-1,1], 129 points.
• gp_d2_level5.txt, Gauss Patterson rule in 2 dimensions, level 5, in [-1,1], 321 points.

SPARSE_GRID_DISPLAY can be used to display images of two dimensional grids, which can be saved in the PNG format:

• gp_d2_level0.png,
• gp_d2_level1.png,
• gp_d2_level2.png,
• gp_d2_level3.png,
• gp_d2_level4.png,
• gp_d2_level5.png,

Gauss-Patterson Sparse Quadrature Rules in 6D, defined on the [-1,1] hypercube:

• gp_d6_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• gp_d6_level0_w.txt, the weights.
• gp_d6_level0_r.txt, the region.
• gp_d6_level1_x.txt, the abscissas for the level 1, order 13 sparse rule.
• gp_d6_level1_w.txt, the weights.
• gp_d6_level1_r.txt, the region.
• gp_d6_level2_x.txt, the abscissas for the level 2, order 97 sparse rule.
• gp_d6_level2_w.txt, the weights.
• gp_d6_level2_r.txt, the region.
• gp_d6_level3_x.txt, the abscissas for the level 3, order 545 sparse rule.
• gp_d6_level3_w.txt, the weights.
• gp_d6_level3_r.txt, the region.
• gp_d6_level4_x.txt, the abscissas for the level 4, order 2561 sparse rule.
• gp_d6_level4_w.txt, the weights.
• gp_d6_level4_r.txt, the region.
• gp_d6_level5_x.txt, the abscissas for the level 5, order 10625 sparse rule.
• gp_d6_level5_w.txt, the weights.
• gp_d6_level5_r.txt, the region.

Gauss-Patterson Sparse Quadrature Rules in 10D, defined on the [-1,1] hypercube:

• gp_d10_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• gp_d10_level0_w.txt, the weights.
• gp_d10_level0_r.txt, the region.
• gp_d10_level1_x.txt, the abscissas for the level 1, order 21 sparse rule.
• gp_d10_level1_w.txt, the weights.
• gp_d10_level1_r.txt, the region.
• gp_d10_level2_x.txt, the abscissas for the level 2, order 241 sparse rule.
• gp_d10_level2_w.txt, the weights.
• gp_d10_level2_r.txt, the region.
• gp_d10_level3_x.txt, the abscissas for the level 3, order 2001 sparse rule.
• gp_d10_level3_w.txt, the weights.
• gp_d10_level3_r.txt, the region.
• gp_d10_level4_x.txt, the abscissas for the level 4, order 13441 sparse rule.
• gp_d10_level4_w.txt, the weights.
• gp_d10_level4_r.txt, the region.
• gp_d10_level5_x.txt, the abscissas for the level 5, order 77505 sparse rule.
• gp_d10_level5_w.txt, the weights.
• gp_d10_level5_r.txt, the region.

You can go up one level to the DATASETS page.