SPARSE_GRID_COMPOSITE
Sparse Grids
Composite Midpoint Rule

SPARSE_GRID_COMPOSITE is a dataset directory which contains examples of multidimensional sparse grid quadrature rules based on the one-dimensional composite midpoint 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 composite midpoint rule, for level 2 which has 13 points.

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

       -0.622222
       -0.088889
       -0.088889
       -0.088889
       -0.088889
        0.800000
        0.800000
        0.444444
        0.444444
        0.444444
        0.444444
        0.800000
        0.800000
      

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

        0.000000        0.000000
       -1.000000        0.000000
        1.000000        0.000000
        0.000000       -1.000000
        0.000000        1.000000
       -0.500000        0.000000
        0.500000        0.000000
       -1.000000       -1.000000
        1.000000       -1.000000
       -1.000000        1.000000
        1.000000        1.000000
        0.000000       -0.500000
        0.000000        0.500000
      

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  
      

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:

NINT_EXACTNESS_MIXED, a C++ program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

SMOLPACK, a C library which estimates the integral of a function over a multidimensional hypercube using a sparse grid, by Knut Petras;

SPARSE_GRID_COMPOSITE, a MATLAB library which creates sparse grids based on 1D composite rules (currently only of order 1).

SPARSE_GRID_DISPLAY, a MATLAB program which reads a file of points on a sparse grid, displays the grid and save the image in a PNG file;

SPARSE_GRID_MIXED, a C++ library which creates a sparse grid dataset based on a mixed set of 1D factor rules.

SPINTERP, a MATLAB library which carries out piecewise multilinear hierarchical sparse grid interpolation, by Andreas Klimke; ACM TOMS Algorithm 847.

TABLE, a data directory which contains examples of TABLE files, a simple format for N points in M dimensions;

Sample Files:

COMPOSITE_D2_LEVEL0 is a composite rule sparse grid of level 0, in dimension 2:

• composite_d2_level0_x.txt, the abscissas.
• composite_d2_level0_w.txt, the weights.
• composite_d2_level0_r.txt, the region.
• composite_d2_level0.png, an image of the grid.

COMPOSITE_D2_LEVEL1 is a composite rule sparse grid of level 1, in dimension 2:

• composite_d2_level1_x.txt, the abscissas.
• composite_d2_level1_w.txt, the weights.
• composite_d2_level1_r.txt, the region.
• composite_d2_level1.png, an image of the grid.

COMPOSITE_D2_LEVEL2 is a composite rule sparse grid of level 2, in dimension 2:

• composite_d2_level2_x.txt, the abscissas.
• composite_d2_level2_w.txt, the weights.
• composite_d2_level2_r.txt, the region.
• composite_d2_level2.png, an image of the grid.

COMPOSITE_D2_LEVEL3 is a composite rule sparse grid of level 3, in dimension 2:

• composite_d2_level3_x.txt, the abscissas.
• composite_d2_level3_w.txt, the weights.
• composite_d2_level3_r.txt, the region.
• composite_d2_level3.png, an image of the grid.

COMPOSITE_D2_LEVEL4 is a composite rule sparse grid of level 4, in dimension 2:

• composite_d2_level4_x.txt, the abscissas.
• composite_d2_level4_w.txt, the weights.
• composite_d2_level4_r.txt, the region.
• composite_d2_level4.png, an image of the grid.

COMPOSITE_D2_LEVEL5 is a composite rule sparse grid of level 5, in dimension 2:

• composite_d2_level5_x.txt, the abscissas.
• composite_d2_level5_w.txt, the weights.
• composite_d2_level5_r.txt, the region.
• composite_d2_level5.png, an image of the grid.

You can go up one level to the DATASETS page.