SPARSE_GRID_NCC - Sparse Grid Files
Newton Cotes Closed

SPARSE_GRID_NCC is a dataset directory which contains 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
      

Licensing:

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

Related Data and Programs:

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.

SPINTERP, a MATLAB library which uses a sparse grid to perform multilinear hierarchical interpolation, by Andreas Klimke

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.