Sparse Grid Files
Fejer Type 2

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 Fejer Type 2 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 Fejer type 2 rule of level 1.

      0.000000      0.000000      0.000000      0.000000      0.000000      0.000000
     -0.707107      0.000000      0.000000      0.000000      0.000000      0.000000
      0.707107      0.000000      0.000000      0.000000      0.000000      0.000000
      0.000000     -0.707107      0.000000      0.000000      0.000000      0.000000
      0.000000      0.707107      0.000000      0.000000      0.000000      0.000000
      0.000000      0.000000     -0.707107      0.000000      0.000000      0.000000
      0.000000      0.000000      0.707107      0.000000      0.000000      0.000000
      0.000000      0.000000      0.000000     -0.707107      0.000000      0.000000
      0.000000      0.000000      0.000000      0.707107      0.000000      0.000000
      0.000000      0.000000      0.000000      0.000000     -0.707107      0.000000
      0.000000      0.000000      0.000000      0.000000      0.707107      0.000000
      0.000000      0.000000      0.000000      0.000000      0.000000     -0.707107
      0.000000      0.000000      0.000000      0.000000      0.000000      0.707107
      

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_GP some sample sparse grids from the Gauss Patterson 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:

• f2_d2_level0.txt, Fejer Type 2 rule in 2 dimensions, level 0, in [-1,1], 1 point.
• f2_d2_level1.txt, Fejer Type 2 rule in 2 dimensions, level 1, in [-1,1], 5 points.
• f2_d2_level2.txt, Fejer Type 2 rule in 2 dimensions, level 2, in [-1,1], 17 points.
• f2_d2_level3.txt, Fejer Type 2 rule in 2 dimensions, level 3, in [-1,1], 49 points.
• f2_d2_level4.txt, Fejer Type 2 rule in 2 dimensions, level 4, in [-1,1], 129 points.
• f2_d2_level5.txt, Fejer Type 2 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:

• f2_d2_level0.png,
• f2_d2_level1.png,
• f2_d2_level2.png,
• f2_d2_level3.png,
• f2_d2_level4.png,
• f2_d2_level5.png,

Some grids in 6D:

• f2_d6_level0.txt, Fejer Type 2 rule in 6 dimensions, level 0, 1 point.
• f2_d6_level1.txt, Fejer Type 2 rule in 6 dimensions, level 1, 13 points.
• f2_d6_level2.txt, Fejer Type 2 rule in 6 dimensions, level 2, 97 points.
• f2_d6_level3.txt, Fejer Type 2 rule in 6 dimensions, level 3, 545 points.

You can go up one level to the DATASETS page.