SPARSE_GRID_F2S
Fejer Type 2 Slow-Growth Sparse Grids

SPARSE_GRID_F2S is a dataset directory which contains examples of multidimensional sparse grid quadrature rules based on the one-dimensional Fejer Type 2 rule with Slow Exponential Growth (F2S)

The slow exponential growth rate refers to the relationship between the 1D level and order of the Fejer Type 2 rules that are used. In the classical F2 rule, the rules have orders of ( 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, ... ) so that the order O = 2^(L+1) - 1, and this is also the value of the precision.

At level L, the slow growth Fejer Type 2 (F2S) rule chooses the classical F2 of lowest order that equals or exceeds 2 * L + 1. Thus, the F2S orders begin with ( 1, 3, 7, 7, 15, 15, 15, 15, 31, 31, 31, ... ).

Particularly for low dimensions, sparse grids made from the F2S family can have significantly fewer points than those made from the F2 family, while maintaining a precision of 2*L+1.

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.

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, a C++ program which measures the polynomial exactness of a multidimensional quadrature rule.

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

SPARSE_GRID_CCS, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Clenshaw Curtis rule with Slow Exponential Growth;

SPARSE_GRID_F2, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Fejer Type 2 rule;

SPARSE_GRID_GPS, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Gauss-Patterson Slow-Growth rule;

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

SPARSE_GRID_MIXED_DATASET, a C++ program which creates a sparse grid dataset based on a mixture of 1D rules.

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

Sample Files:

F2S rules in 2D.

• f2s_d2_level0_x.txt, the abscissas.
• f2s_d2_level0_w.txt, the weights.
• f2s_d2_level0_r.txt, the region.
• f2s_d2_level0.png, an image.
• f2s_d2_level1_x.txt, the abscissas.
• f2s_d2_level1_w.txt, the weights.
• f2s_d2_level1_r.txt, the region.
• f2s_d2_level1.png, an image.
• f2s_d2_level2_x.txt, the abscissas.
• f2s_d2_level2_w.txt, the weights.
• f2s_d2_level2_r.txt, the region.
• f2s_d2_level2.png, an image.
• f2s_d2_level3_x.txt, the abscissas.
• f2s_d2_level3_w.txt, the weights.
• f2s_d2_level3_r.txt, the region.
• f2s_d2_level3.png, an image.
• f2s_d2_level4_x.txt, the abscissas.
• f2s_d2_level4_w.txt, the weights.
• f2s_d2_level4_r.txt, the region.
• f2s_d2_level4.png, an image.
• f2s_d2_level5_x.txt, the abscissas.
• f2s_d2_level5_w.txt, the weights.
• f2s_d2_level5_r.txt, the region.
• f2s_d2_level5.png, an image.
• f2s_d2_level6_x.txt, the abscissas.
• f2s_d2_level6_w.txt, the weights.
• f2s_d2_level6_r.txt, the region.
• f2s_d2_level6.png, an image.
• f2s_d2_level7_x.txt, the abscissas.
• f2s_d2_level7_w.txt, the weights.
• f2s_d2_level7_r.txt, the region.
• f2s_d2_level7.png, an image.
• f2s_d2_level8_x.txt, the abscissas.
• f2s_d2_level8_w.txt, the weights.
• f2s_d2_level8_r.txt, the region.
• f2s_d2_level8.png, an image.
• f2s_d2_level9_x.txt, the abscissas.
• f2s_d2_level9_w.txt, the weights.
• f2s_d2_level9_r.txt, the region.
• f2s_d2_level9.png, an image.
• f2s_d2_level10_x.txt, the abscissas.
• f2s_d2_level10_w.txt, the weights.
• f2s_d2_level10_r.txt, the region.
• f2s_d2_level10.png, an image.

F2S rules in 3D:

• f2s_d3_level0_x.txt, the abscissas.
• f2s_d3_level0_w.txt, the weights.
• f2s_d3_level0_r.txt, the region.
• f2s_d3_level0.png, an image.
• f2s_d3_level1_x.txt, the abscissas.
• f2s_d3_level1_w.txt, the weights.
• f2s_d3_level1_r.txt, the region.
• f2s_d3_level1.png, an image.
• f2s_d3_level2_x.txt, the abscissas.
• f2s_d3_level2_w.txt, the weights.
• f2s_d3_level2_r.txt, the region.
• f2s_d3_level2.png, an image.
• f2s_d3_level3_x.txt, the abscissas.
• f2s_d3_level3_w.txt, the weights.
• f2s_d3_level3_r.txt, the region.
• f2s_d3_level3.png, an image.
• f2s_d3_level4_x.txt, the abscissas.
• f2s_d3_level4_w.txt, the weights.
• f2s_d3_level4_r.txt, the region.
• f2s_d3_level4.png, an image.
• f2s_d3_level5_x.txt, the abscissas.
• f2s_d3_level5_w.txt, the weights.
• f2s_d3_level5_r.txt, the region.
• f2s_d3_level5.png, an image.
• f2s_d3_level6_x.txt, the abscissas.
• f2s_d3_level6_w.txt, the weights.
• f2s_d3_level6_r.txt, the region.
• f2s_d3_level6.png, an image.
• f2s_d3_level7_x.txt, the abscissas.
• f2s_d3_level7_w.txt, the weights.
• f2s_d3_level7_r.txt, the region.
• f2s_d3_level7.png, an image.
• f2s_d3_level8_x.txt, the abscissas.
• f2s_d3_level8_w.txt, the weights.
• f2s_d3_level8_r.txt, the region.
• f2s_d3_level9_x.txt, the abscissas.
• f2s_d3_level9_w.txt, the weights.
• f2s_d3_level9_r.txt, the region.
• f2s_d3_level10_x.txt, the abscissas.
• f2s_d3_level10_w.txt, the weights.
• f2s_d3_level10_r.txt, the region.

You can go up one level to the DATASETS page.