Sparse Grids
Clenshaw Curtis

These are some examples of multidimensional sparse grid quadrature rules based on the one-dimensional Clenshaw Curtis rule.

A quadrature rule is a set of n points x and associated weights w so that the integral of a function f(x) over some particular region can be approximated by:

Integral f(x) dx = Sum ( 1 <= i <= n ) w(i) * f(x(i))

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the corners of the rectangular region. The dimension of the region is deduced implicitly from the dimension of the points.

Example:

Here are the three files associated with a sparse grid for two dimensions, based on the Clenshaw Curtis rule, for level 2 which has 13 points.

Here is the text of an "W" file storing the weights of such a rule

     -0.3555555555555558
     -0.0888888888888890
     -0.0888888888888890
     -0.0888888888888889
     -0.0888888888888889
      1.0666666666666667
      1.0666666666666669
      0.1111111111111111
      0.1111111111111111
      0.1111111111111111
      0.1111111111111111
      1.0666666666666667
      1.0666666666666669
      

Here is the text of a "X" file storing the abscissas of such a rule:

      0.0000000000000001      0.0000000000000001
     -1.0000000000000000      0.0000000000000001
      1.0000000000000000      0.0000000000000001
      0.0000000000000001     -1.0000000000000000
      0.0000000000000001      1.0000000000000000
     -0.7071067811865475      0.0000000000000001
      0.7071067811865476      0.0000000000000001
     -1.0000000000000000     -1.0000000000000000
     -1.0000000000000000      1.0000000000000000
      1.0000000000000000     -1.0000000000000000
      1.0000000000000000      1.0000000000000000
      0.0000000000000001     -0.7071067811865475
      0.0000000000000001      0.7071067811865476
      

Here is the text of an "R" file storing the lower and upper limits of the region, needed to determine the integration region:

    -1.0000000000000000        -1.0000000000000000    
     1.0000000000000000         1.0000000000000000  
      

Related Data and Programs:

• CLENSHAW_CURTIS is a FORTRAN90 library of routines which can compute Clenshaw Curtis grids.
• 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;

Other collections of quadrature rules:

• 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_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_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 1D:

• cc_d1_level0_x.txt, the abscissas for the level 0, order 1 rule.
• cc_d1_level0_w.txt, the weightse.
• cc_d1_level0_r.txt, the range.
• cc_d1_level1_x.txt, the abscissas for the level 1, order 3 rule.
• cc_d1_level1_w.txt, the weights.
• cc_d1_level1_r.txt, the range.
• cc_d1_level2_x.txt, the abscissas for the level 2, order 5 rule.
• cc_d1_level2_w.txt, the weightse.
• cc_d1_level2_r.txt, the range.
• cc_d1_level3_x.txt, the abscissas for the level 3, order 9 rule.
• cc_d1_level3_w.txt, the weights.
• cc_d1_level3_r.txt, the range.
• cc_d1_level4_x.txt, the abscissas for the level 4, order 17 rule.
• cc_d1_level4_w.txt, the weightse.
• cc_d1_level4_r.txt, the range.
• cc_d1_level5_x.txt, the abscissas for the level 5, order 33 rule.
• cc_d1_level5_w.txt, the weights.
• cc_d1_level5_r.txt, the range.
• cc_d1_level6_x.txt, the abscissas for the level 6, order 65 rule.
• cc_d1_level6_w.txt, the weights.
• cc_d1_level6_r.txt, the range.

Clenshaw Curtis Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• cc_d2_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d2_level0_w.txt, the weights.
• cc_d2_level0_r.txt, the region.
• cc_d2_level1_x.txt, the abscissas for the level 1, order 5 sparse rule.
• cc_d2_level1_w.txt, the weights.
• cc_d2_level1_r.txt, the region.
• cc_d2_level2_x.txt, the abscissas for the level 2, order 13 sparse rule.
• cc_d2_level2_w.txt, the weights.
• cc_d2_level2_r.txt, the region.
• cc_d2_level3_x.txt, the abscissas for the level 3, order 29 sparse rule.
• cc_d2_level3_w.txt, the weights.
• cc_d2_level3_r.txt, the region.
• cc_d2_level4_x.txt, the abscissas for the level 4, order 65 sparse rule.
• cc_d2_level4_w.txt, the weights.
• cc_d2_level4_r.txt, the region.
• cc_d2_level5_x.txt, the abscissas for the level 5, order 145 sparse rule.
• cc_d2_level5_w.txt, the weights.
• cc_d2_level5_r.txt, the region.
• cc_d2_level6_x.txt, the abscissas for the level 6, order 321 sparse rule.
• cc_d2_level6_w.txt, the weights.
• cc_d2_level6_r.txt, the region.
• cc_d2_level7_x.txt, the abscissas for the level 7, order 705 sparse rule.
• cc_d2_level7_w.txt, the weights.
• cc_d2_level7_r.txt, the region.
• cc_d2_level8_x.txt, the abscissas for the level 8, order 1537 sparse rule.
• cc_d2_level8_w.txt, the weights.
• cc_d2_level8_r.txt, the region.
• cc_d2_level9_x.txt, the abscissas for the level 9, order 3329 sparse rule.
• cc_d2_level9_w.txt, the weights.
• cc_d2_level9_r.txt, the region.
• cc_d2_level10_x.txt, the abscissas for the level 10, order 7169 sparse rule.
• cc_d2_level10_w.txt, the weights.
• cc_d2_level10_r.txt, the region.

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

• cc_d2_level0_x.png,
• cc_d2_level1_x.png,
• cc_d2_level2_x.png,
• cc_d2_level3_x.png,
• cc_d2_level4_x.png,
• cc_d2_level5_x.png,
• cc_d2_level6_x.png,

Clenshaw Curtis Sparse Quadrature Rules in 3D, defined on the [-1,1] cube:

• cc_d3_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d3_level0_w.txt, the weights.
• cc_d3_level0_r.txt, the region.
• cc_d3_level1_x.txt, the abscissas for the level 1, order 7 sparse rule.
• cc_d3_level1_w.txt, the weights.
• cc_d3_level1_r.txt, the region.
• cc_d3_level2_x.txt, the abscissas for the level 2, order 25 sparse rule.
• cc_d3_level2_w.txt, the weights.
• cc_d3_level2_r.txt, the region.
• cc_d3_level3_x.txt, the abscissas for the level 3, order 69 sparse rule.
• cc_d3_level3_w.txt, the weights.
• cc_d3_level3_r.txt, the region.
• cc_d3_level4_x.txt, the abscissas for the level 4, order 177 sparse rule.
• cc_d3_level4_w.txt, the weights.
• cc_d3_level4_r.txt, the region.
• cc_d3_level5_x.txt, the abscissas for the level 5, order 441 sparse rule.
• cc_d3_level5_w.txt, the weights.
• cc_d3_level5_r.txt, the region.
• cc_d3_level6_x.txt, the abscissas for the level 6, order 1073 sparse rule.
• cc_d3_level6_w.txt, the weights.
• cc_d3_level6_r.txt, the region.
• cc_d3_level7_x.txt, the abscissas for the level 7, order 2561 sparse rule.
• cc_d3_level7_w.txt, the weights.
• cc_d3_level7_r.txt, the region.

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

• cc_d3_level0_x.png,
• cc_d3_level1_x.png,
• cc_d3_level2_x.png,
• cc_d3_level3_x.png,
• cc_d3_level4_x.png,
• cc_d3_level5_x.png,
• cc_d3_level6_x.png,

Clenshaw Curtis Sparse Quadrature Rules in 4D, defined on the [-1,1] hypercube:

• cc_d4_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d4_level0_w.txt, the weights.
• cc_d4_level0_r.txt, the region.
• cc_d4_level1_x.txt, the abscissas for the level 1, order 9 sparse rule.
• cc_d4_level1_w.txt, the weights.
• cc_d4_level1_r.txt, the region.
• cc_d4_level2_x.txt, the abscissas for the level 2, order 41 sparse rule.
• cc_d4_level2_w.txt, the weights.
• cc_d4_level2_r.txt, the region.
• cc_d4_level3_x.txt, the abscissas for the level 3, order 137 sparse rule.
• cc_d4_level3_w.txt, the weights.
• cc_d4_level3_r.txt, the region.
• cc_d4_level4_x.txt, the abscissas for the level 4, order 401 sparse rule.
• cc_d4_level4_w.txt, the weights.
• cc_d4_level4_r.txt, the region.
• cc_d4_level5_x.txt, the abscissas for the level 5, order 1105 sparse rule.
• cc_d4_level5_w.txt, the weights.
• cc_d4_level5_r.txt, the region.
• cc_d4_level6_x.txt, the abscissas for the level 6, order 2929 sparse rule.
• cc_d4_level6_w.txt, the weights.
• cc_d4_level6_r.txt, the region.

Clenshaw Curtis Sparse Quadrature Rules in 5D, defined on the [-1,1] hypercube:

• cc_d5_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d5_level0_w.txt, the weights.
• cc_d5_level0_r.txt, the region.
• cc_d5_level1_x.txt, the abscissas for the level 1, order 11 sparse rule.
• cc_d5_level1_w.txt, the weights.
• cc_d5_level1_r.txt, the region.
• cc_d5_level2_x.txt, the abscissas for the level 2, order 61 sparse rule.
• cc_d5_level2_w.txt, the weights.
• cc_d5_level2_r.txt, the region.
• cc_d5_level3_x.txt, the abscissas for the level 3, order 241 sparse rule.
• cc_d5_level3_w.txt, the weights.
• cc_d5_level3_r.txt, the region.
• cc_d5_level4_x.txt, the abscissas for the level 4, order 801 sparse rule.
• cc_d5_level4_w.txt, the weights.
• cc_d5_level4_r.txt, the region.
• cc_d5_level5_x.txt, the abscissas for the level 5, order 2433 sparse rule.
• cc_d5_level5_w.txt, the weights.
• cc_d5_level5_r.txt, the region.

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

• cc_d6_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d6_level0_w.txt, the weights.
• cc_d6_level0_r.txt, the region.
• cc_d6_level1_x.txt, the abscissas for the level 1, order 13 sparse rule.
• cc_d6_level1_w.txt, the weights.
• cc_d6_level1_r.txt, the region.
• cc_d6_level2_x.txt, the abscissas for the level 2, order 85 sparse rule.
• cc_d6_level2_w.txt, the weights.
• cc_d6_level2_r.txt, the region.
• cc_d6_level3_x.txt, the abscissas for the level 3, order 389 sparse rule.
• cc_d6_level3_w.txt, the weights.
• cc_d6_level3_r.txt, the region.
• cc_d6_level4_x.txt, the abscissas for the level 4, order 1457 sparse rule.
• cc_d6_level4_w.txt, the weights.
• cc_d6_level4_r.txt, the region.
• cc_d6_level5_x.txt, the abscissas for the level 5, order 4865 sparse rule.
• cc_d6_level5_w.txt, the weights.
• cc_d6_level5_r.txt, the region.

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

• cc_d10_level0_x.txt, the abscissas for the level 0, order 1 sparse rule.
• cc_d10_level0_w.txt, the weights.
• cc_d10_level0_r.txt, the region.
• cc_d10_level1_x.txt, the abscissas for the level 1, order 21 sparse rule.
• cc_d10_level1_w.txt, the weights.
• cc_d10_level1_r.txt, the region.
• cc_d10_level2_x.txt, the abscissas for the level 2, order 221 sparse rule.
• cc_d10_level2_w.txt, the weights.
• cc_d10_level2_r.txt, the region.
• cc_d10_level3_x.txt, the abscissas for the level 3, order 1581 sparse rule.
• cc_d10_level3_w.txt, the weights.
• cc_d10_level3_r.txt, the region.
• cc_d10_level4_x.txt, the abscissas for the level 4, order 8801 sparse rule.
• cc_d10_level4_w.txt, the weights.
• cc_d10_level4_r.txt, the region.
• cc_d10_level5_x.txt, the abscissas for the level 5, order 41265 sparse rule.
• cc_d10_level5_w.txt, the weights.
• cc_d10_level5_r.txt, the region.

You can go up one level to the DATASETS page.