SPARSE_GRID_CCL
Clenshaw Curtis Linear-Growth Sparse Grids

SPARSE_GRID_CCL is a dataset directory which contains examples of multidimensional sparse grid quadrature rules based on the one-dimensional Clenshaw Curtis rule with Linear Growth (CCL)

The linear growth rate refers to the relationship between the 1D level and order of the Clenshaw Curtis rules that are used. In the classical CC rule, the rules have orders of ( 1, 3, 5, 9, 17, 33, ... ) so that for 0 < L, the order O = 2^L+1.

At level L, the linear growth Clenshaw Curtis (CCL) rule chooses the classical Clenshaw of order O = 2 * L + 1. Thus, the CCL orders begin with ( 1, 3, 5, 7, 9, 11, 13, ... ).

It is interesting to compare sparse grids constructed from the CCL and CCS rules, both of which attempt to preserve the precision property of classical CC sparse grids, while trying to reduce the rate of point growth.

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_CCE, a dataset directory which contains multidimensional Smolyak sparse grids based on the Clenshaw Curtis Exponential growth rule;

SPARSE_GRID_CCS, a dataset directory which contains multidimensional Smolyak sparse grids based on the Clenshaw Curtis Slow growth rule;

Sample Files:

CCL rules in 1D:

• ccl_d1_level0_x.txt, the 1 abscissa.
• ccl_d1_level0_w.txt, the weights.
• ccl_d1_level0_r.txt, the region.
• ccl_d1_level1_x.txt, the 3 abscissas.
• ccl_d1_level1_w.txt, the weights.
• ccl_d1_level1_r.txt, the region.
• ccl_d1_level2_x.txt, the 5 abscissas.
• ccl_d1_level2_w.txt, the weights.
• ccl_d1_level2_r.txt, the region.
• ccl_d1_level3_x.txt, the 7 abscissas.
• ccl_d1_level3_w.txt, the weights.
• ccl_d1_level3_r.txt, the region.
• ccl_d1_level4_x.txt, the 9 abscissas.
• ccl_d1_level4_w.txt, the weights.
• ccl_d1_level4_r.txt, the region.
• ccl_d1_level5_x.txt, the 11 abscissas.
• ccl_d1_level5_w.txt, the weights.
• ccl_d1_level5_r.txt, the region.
• ccl_d1_level6_x.txt, the 13 abscissas.
• ccl_d1_level6_w.txt, the weights.
• ccl_d1_level6_r.txt, the region.
• ccl_d1_level7_x.txt, the 15 abscissas.
• ccl_d1_level7_w.txt, the weights.
• ccl_d1_level7_r.txt, the region.
• ccl_d1_level8_x.txt, the 17 abscissas.
• ccl_d1_level8_w.txt, the weights.
• ccl_d1_level8_r.txt, the region.
• ccl_d1_level9_x.txt, the 19 abscissas.
• ccl_d1_level9_w.txt, the weights.
• ccl_d1_level9_r.txt, the region.

CCL rules in 2D:

• ccl_d2_level0_x.txt, the 1 abscissa.
• ccl_d2_level0_w.txt, the weights.
• ccl_d2_level0_r.txt, the region.
• ccl_d2_level1_x.txt, the 7 abscissas.
• ccl_d2_level1_w.txt, the weights.
• ccl_d2_level1_r.txt, the region.
• ccl_d2_level2_x.txt, the 25 abscissas.
• ccl_d2_level2_w.txt, the weights.
• ccl_d2_level2_r.txt, the region.
• ccl_d2_level3_x.txt, the 63 abscissas.
• ccl_d2_level3_w.txt, the weights.
• ccl_d2_level3_r.txt, the region.
• ccl_d2_level4_x.txt, the 129 abscissas.
• ccl_d2_level4_w.txt, the weights.
• ccl_d2_level4_r.txt, the region.
• ccl_d2_level5_x.txt, the 231 abscissas.
• ccl_d2_level5_w.txt, the weights.
• ccl_d2_level5_r.txt, the region.
• ccl_d2_level6_x.txt, the 377 abscissas.
• ccl_d2_level6_w.txt, the weights.
• ccl_d2_level6_r.txt, the region.
• ccl_d2_level7_x.txt, the 575 abscissas.
• ccl_d2_level7_w.txt, the weights.
• ccl_d2_level7_r.txt, the region.
• ccl_d2_level8_x.txt, the 833 abscissas.
• ccl_d2_level8_w.txt, the weights.
• ccl_d2_level8_r.txt, the region.
• ccl_d2_level9_x.txt, the 1159 abscissas.
• ccl_d2_level9_w.txt, the weights.
• ccl_d2_level9_r.txt, the region.
• ccl_d2_level10_x.txt, the 1561 abscissas.
• ccl_d2_level10_w.txt, the weights.
• ccl_d2_level10_r.txt, the region.

CCS rules in 6D:

• ccl_d6_level0_x.txt, the 1 abscissa.
• ccl_d6_level0_w.txt, the weights.
• ccl_d6_level0_r.txt, the region.
• ccl_d6_level1_x.txt, the 19 abscissas.
• ccl_d6_level1_w.txt, the weights.
• ccl_d6_level1_r.txt, the region.
• ccl_d6_level2_x.txt, the 184 abscissas.
• ccl_d6_level2_w.txt, the weights.
• ccl_d6_level2_r.txt, the region.
• ccl_d6_level3_x.txt, the 1216 abscissas.
• ccl_d6_level3_w.txt, the weights.
• ccl_d6_level3_r.txt, the region.
• ccl_d6_level4_x.txt, the 6190 abscissas.
• ccl_d6_level4_w.txt, the weights.
• ccl_d6_level4_r.txt, the region.
• ccl_d6_level5_x.txt, the 25,954 abscissas.
• ccl_d6_level5_w.txt, the weights.
• ccl_d6_level5_r.txt, the region.
• ccl_d6_level6_x.txt, the 93,535 abscissas.
• ccl_d6_level6_w.txt, the weights.
• ccl_d6_level6_r.txt, the region.

You can go up one level to the DATASETS page.