Sparse Grid Files
Newton Cotes Closed

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 Newton Cotes Closed 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 Newton Cotes Closed rule of level 2.

      0.500000      0.500000
      0.000000      0.500000
      1.000000      0.500000
      0.500000      0.000000
      0.500000      1.000000
      0.250000      0.500000
      0.750000      0.500000
      0.000000      0.000000
      0.000000      1.000000
      1.000000      0.000000
      1.000000      1.000000
      0.500000      0.250000
      0.500000      0.750000

      

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_GP some sample sparse grids from the Gauss Patterson 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:

• ncc_d2_level0.txt, Newton Cotes Closed rule in 2 dimensions, level 0, in [0,1], 1 point.
• ncc_d2_level1.txt, Newton Cotes Closed rule in 2 dimensions, level 1, in [0,1], 5 points.
• ncc_d2_level2.txt, Newton Cotes Closed rule in 2 dimensions, level 2, in [0,1], 13 points.
• ncc_d2_level3.txt, Newton Cotes Closed rule in 2 dimensions, level 3, in [0,1], 29 points.
• ncc_d2_level4.txt, Newton Cotes Closed rule in 2 dimensions, level 4, in [0,1], 65 points.
• ncc_d2_level5.txt, Newton Cotes Closed rule in 2 dimensions, level 5, in [0,1], 145 points.
• ncc_d2_level6.txt, Newton Cotes Closed rule in 2 dimensions, level 6, in [0,1], 321 points.

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

• ncc_d2_level0.png,
• ncc_d2_level1.png,
• ncc_d2_level2.png,
• ncc_d2_level3.png,
• ncc_d2_level4.png,
• ncc_d2_level5.png,
• ncc_d2_level6.png,

Some grids in 3D:

• ncc_d3_level0.txt, Newton Cotes Closed rule in 3 dimensions, level 0, in [0,1], 1 point.
• ncc_d3_level1.txt, Newton Cotes Closed rule in 3 dimensions, level 1, in [0,1], 7 points.
• ncc_d3_level2.txt, Newton Cotes Closed rule in 3 dimensions, level 2, in [0,1], 25 points.
• ncc_d3_level3.txt, Newton Cotes Closed rule in 3 dimensions, level 3, in [0,1], 69 points.
• ncc_d3_level4.txt, Newton Cotes Closed rule in 3 dimensions, level 4, in [0,1], 177 points.
• ncc_d3_level5.txt, Newton Cotes Closed rule in 3 dimensions, level 5, in [0,1], 441 points.
• ncc_d3_level6.txt, Newton Cotes Closed rule in 3 dimensions, level 6, in [0,1], 1073 points.

SPARSE_GRID_DISPLAY can be used to display images of 3D grids, which can be saved in the PNG format:

• ncc_d3_level0.png,
• ncc_d3_level1.png,
• ncc_d3_level2.png,
• ncc_d3_level3.png,
• ncc_d3_level4.png,
• ncc_d3_level5.png,
• ncc_d3_level6.png,

Newton Cotes Closed Sparse Quadrature Rules in 6D, defined on the [-1,1] hypercube:

• ncc_d6_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• ncc_d6_level0_w.txt, the weights.
• ncc_d6_level0_r.txt, the region.
• ncc_d6_level1_x.txt, the abscissas for the level 1, order 13 sparse rule.
• ncc_d6_level1_w.txt, the weights.
• ncc_d6_level1_r.txt, the region.
• ncc_d6_level2_x.txt, the abscissas for the level 2, order 85 sparse rule.
• ncc_d6_level2_w.txt, the weights.
• ncc_d6_level2_r.txt, the region.
• ncc_d6_level3_x.txt, the abscissas for the level 3, order 389 sparse rule.
• ncc_d6_level3_w.txt, the weights.
• ncc_d6_level3_r.txt, the region.
• ncc_d6_level4_x.txt, the abscissas for the level 4, order 1457 sparse rule.
• ncc_d6_level4_w.txt, the weights.
• ncc_d6_level4_r.txt, the region.
• ncc_d6_level5_x.txt, the abscissas for the level 5, order 4865 sparse rule.
• ncc_d6_level5_w.txt, the weights. WARNING: these weights had to be computed with QUADRUPLE ACCURACY; otherwise, the computation lost all accuracy.
• ncc_d6_level5_r.txt, the region.

Newton Cotes Closed Sparse Quadrature Rules in 10D, defined on the [-1,1] hypercube:

• ncc_d10_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• ncc_d10_level0_w.txt, the weights.
• ncc_d10_level0_r.txt, the region.
• ncc_d10_level1_x.txt, the abscissas for the level 1, order 21 sparse rule.
• ncc_d10_level1_w.txt, the weights.
• ncc_d10_level1_r.txt, the region.
• ncc_d10_level2_x.txt, the abscissas for the level 2, order 221 sparse rule.
• ncc_d10_level2_w.txt, the weights.
• ncc_d10_level2_r.txt, the region.
• ncc_d10_level3_x.txt, the abscissas for the level 3, order 1581 sparse rule.
• ncc_d10_level3_w.txt, the weights.
• ncc_d10_level3_r.txt, the region.
• ncc_d10_level4_x.txt, the abscissas for the level 4, order 8801 sparse rule.
• ncc_d10_level4_w.txt, the weights.
• ncc_d10_level4_r.txt, the region.
• ncc_d10_level5_x.txt, the abscissas for the level 5, order 41265 sparse rule.
• ncc_d10_level5_w.txt, the weights. WARNING: these weights had to be computed with QUADRUPLE ACCURACY; otherwise, the computation lost all accuracy.
• ncc_d10_level5_r.txt, the region.

You can go up one level to the DATASETS page.