SPARSE_GRID_GPS
Gauss Patterson Slow growth Sparse Grids

SPARSE_GRID_GPS is a dataset directory which contains examples of sparse grid rules based on the Gauss Patterson Slow growth rule.

The slow exponential growth rate refers to the relationship between the 1D level and order of the Gauss Patterson rules that are used. In the classical GP rule, the rules have orders of ( 1, 3, 7, 15, 31, 63, ... ) so that the order O = 2^(L+1) - 1. Meanwhile, these rules have precision of P = 3 * 2^L-1.

At level L, the slow growth Gauss Patterson (GPS) rule chooses the classical GP of lowest order that equals or exceeds 2 * L + 1. Thus, the GPS orders begin with ( 1, 3, 3, 7, 7, 7, 15, 15, 15, 15, 15, ... ).

Particularly for low dimensions, sparse grids made from the GPS family can have significantly fewer points than those made from the GP 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:

SPARSE_GRID_GPE, a dataset directory which contains multidimensional Smolyak sparse grids based on the Gauss-Patterson Exponential growth rule;

Sample Files:

GPS rules in 2D. Notice that the rules of level 4 and 5, of 7 and 8, and of 9 and 10 are equal. This means that in this low dimension, the Gauss Patterson Slow-Growth sparse grids don't mesh perfectly with our requested precision formulas.

• gps_d2_level0_x.txt, the abscissas.
• gps_d2_level0_w.txt, the weights.
• gps_d2_level0_r.txt, the region.
• gps_d2_level0.png, an image.
• gps_d2_level1_x.txt, the abscissas.
• gps_d2_level1_w.txt, the weights.
• gps_d2_level1_r.txt, the region.
• gps_d2_level1.png, an image.
• gps_d2_level2_x.txt, the abscissas.
• gps_d2_level2_w.txt, the weights.
• gps_d2_level2_r.txt, the region.
• gps_d2_level2.png, an image.
• gps_d2_level3_x.txt, the abscissas.
• gps_d2_level3_w.txt, the weights.
• gps_d2_level3_r.txt, the region.
• gps_d2_level3.png, an image.
• gps_d2_level4_x.txt, the abscissas.
• gps_d2_level4_w.txt, the weights.
• gps_d2_level4_r.txt, the region.
• gps_d2_level4.png, an image.
• gps_d2_level5_x.txt, the abscissas.
• gps_d2_level5_w.txt, the weights.
• gps_d2_level5_r.txt, the region.
• gps_d2_level5.png, an image.
• gps_d2_level6_x.txt, the abscissas.
• gps_d2_level6_w.txt, the weights.
• gps_d2_level6_r.txt, the region.
• gps_d2_level6.png, an image.
• gps_d2_level7_x.txt, the abscissas.
• gps_d2_level7_w.txt, the weights.
• gps_d2_level7_r.txt, the region.
• gps_d2_level7.png, an image.
• gps_d2_level8_x.txt, the abscissas.
• gps_d2_level8_w.txt, the weights.
• gps_d2_level8_r.txt, the region.
• gps_d2_level8.png, an image.
• gps_d2_level9_x.txt, the abscissas.
• gps_d2_level9_w.txt, the weights.
• gps_d2_level9_r.txt, the region.
• gps_d2_level9.png, an image.
• gps_d2_level10_x.txt, the abscissas.
• gps_d2_level10_w.txt, the weights.
• gps_d2_level10_r.txt, the region.
• gps_d2_level10.png, an image.

GPS rules in 3D:

• gps_d3_level0_x.txt, the abscissas.
• gps_d3_level0_w.txt, the weights.
• gps_d3_level0_r.txt, the region.
• gps_d3_level0.png, an image.
• gps_d3_level1_x.txt, the abscissas.
• gps_d3_level1_w.txt, the weights.
• gps_d3_level1_r.txt, the region.
• gps_d3_level1.png, an image.
• gps_d3_level2_x.txt, the abscissas.
• gps_d3_level2_w.txt, the weights.
• gps_d3_level2_r.txt, the region.
• gps_d3_level2.png, an image.
• gps_d3_level3_x.txt, the abscissas.
• gps_d3_level3_w.txt, the weights.
• gps_d3_level3_r.txt, the region.
• gps_d3_level3.png, an image.
• gps_d3_level4_x.txt, the abscissas.
• gps_d3_level4_w.txt, the weights.
• gps_d3_level4_r.txt, the region.
• gps_d3_level4.png, an image.
• gps_d3_level5_x.txt, the abscissas.
• gps_d3_level5_w.txt, the weights.
• gps_d3_level5_r.txt, the region.
• gps_d3_level5.png, an image.
• gps_d3_level6_x.txt, the abscissas.
• gps_d3_level6_w.txt, the weights.
• gps_d3_level6_r.txt, the region.
• gps_d3_level6.png, an image.
• gps_d3_level7_x.txt, the abscissas.
• gps_d3_level7_w.txt, the weights.
• gps_d3_level7_r.txt, the region.
• gps_d3_level7.png, an image.
• gps_d3_level8_x.txt, the abscissas.
• gps_d3_level8_w.txt, the weights.
• gps_d3_level8_r.txt, the region.
• gps_d3_level9_x.txt, the abscissas.
• gps_d3_level9_w.txt, the weights.
• gps_d3_level9_r.txt, the region.
• gps_d3_level10_x.txt, the abscissas.
• gps_d3_level10_w.txt, the weights.
• gps_d3_level10_r.txt, the region.

GPS rules in 6D:

• gps_d6_level0_x.txt, the abscissas.
• gps_d6_level0_w.txt, the weights.
• gps_d6_level0_r.txt, the region.
• gps_d6_level1_x.txt, the abscissas.
• gps_d6_level1_w.txt, the weights.
• gps_d6_level1_r.txt, the region.
• gps_d6_level2_x.txt, the abscissas.
• gps_d6_level2_w.txt, the weights.
• gps_d6_level2_r.txt, the region.
• gps_d6_level3_x.txt, the abscissas.
• gps_d6_level3_w.txt, the weights.
• gps_d6_level3_r.txt, the region.
• gps_d6_level4_x.txt, the abscissas.
• gps_d6_level4_w.txt, the weights.
• gps_d6_level4_r.txt, the region.
• gps_d6_level5_x.txt, the abscissas.
• gps_d6_level5_w.txt, the weights.
• gps_d6_level5_r.txt, the region.
• gps_d6_level6_x.txt, the abscissas.
• gps_d6_level6_w.txt, the weights.
• gps_d6_level6_r.txt, the region.

You can go up one level to the DATASETS page.