SGMGA Sparse Grids, Mixed Rules, Growth Rate, Anisotropic Weights

SGMGA is a dataset directory which contains some examples of sparse grids in which different 1D quadrature rules may be used for each spatial dimension, the growth rate for each 1D family is user-specified, and there are anistropic weights specified for each dimension.

The default growth rate depends on the rule. Thus:

• exponential growth rate is reserved, by default, for "CC", "F2" and "GP" rules. linear growth rate for other rules (
• linear growth rate of 2*L+1, where L is the level, is reserved for "GL", "GH", "GGH", "LG", "GLG", "GJ" and "GW" rules.

The following list includes 1D quadrature rules that may be specified as factors. A few rules are not yet supported.

• CC, Clenshaw-Curtis:
defined on [-1,+1], with w(x)=1,
a closed, fully nested rule.
• F2, Fejer Type 2:
defined on (-1,+1), with w(x)=1,
an open, fully nested rule.
• GP, Gauss Patterson:
defined on (-1,+1), with w(x)=1,
a family of the midpoint rule, the 3 point Gauss Legendre rule, and then successive Patterson refinements,
an open, fully nested rule.
• GL, Gauss Legendre:
defined on (-1,+1), with w(x)=1,
an open, weakly nested rule.
• GH, Gauss Hermite:
defined on (-oo,+oo), with w(x)=exp(-x*x),
an open, weakly nested rule.
• GGH, Generalized Gauss Hermite:
defined on (-oo,+oo), with w(x)=|x|^alpha * exp(-x*x),
an open, weakly nested rule.
• LG, Gauss Laguerre:
defined on (0,+oo) with w(x)=exp(-x),
an open, non-nested rule.
• GLG, Generalized Gauss Laguerre:
defined on (0,+oo) with w(x)=x^alpha * exp(-x),
an open, non-nested rule.
• GJ, Gauss Jacobi:
defined on [-1,+1] with w(x)=(1-x)^alpha (1+x)^beta
an open, non-nested rule.
• GW, Golub Welsch:
a rule defined by the user based on the Golub-Welsch algorithm,
with points and weights supplied by external routines,
presumably an open, non-nested rule.

Each sparse grid is stored using the "weighted quadrature rule" format, that is, as 6 files:

• an "A" or "anisotropy" file, which lists the anisotropic weights for each dimension;
• an "N" or "number of parameters" file, which lists the number of parameters (such as ALPHA and BETA) for each dimension;
• an "P" or "parameter" file, which lists all the parameter values in one list;
• an "R" or "region" file, which lists two points that bound the region;
• a "W" or "weight" file, which lists the weight for each abscissa;
• an "X" or "abscissa" file, which lists the abscissas of the rule.

The abscissas are ordered to respect the natural nesting of the sparse grids by level. That is, the file of level 3 points begins by listing the points in the level 2 grid.

Example:

The "R" file stores two points that define the "corners" of the region. Here is the R file for a 2D rule based on 1D rules that both are defined in [-1,+1].

-1.000000     -1.000000
1.000000     +1.000000

In cases where one or both endpoints are infinite, the infinite value is suggested by using the value 1.0E+30.

Here is the text of a "W" file, which stores the weights of rule:

-0.444444
1.111111
1.111111
1.111111
1.111111

Here is the text of an "X" file storing the abscissas of a rule of dimension 2:

0.000000      0.000000
-0.774596      0.000000
0.774596      0.000000
0.000000     -0.774596
0.000000      0.774596

Related Data and Programs:

GRID_DISPLAY, a MATLAB program which reads a file of points on a grid, displays the grid and saves the image in a PNG file;

GRIDS_DISPLAY, a MATLAB program which reads two files of grids, displays the first with hollow blue dots, the second with red dots.

QUADRATURE_RULES, a dataset directory which contains quadrature rules for 1D intervals, 2D rectangles or multidimensional rectangular regions, stored as a file of abscissas, a file of weights, and a file of region limits.

SGMGA, a C++ library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SGMGA, a FORTRAN90 library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SPARSE_GRID_CC, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Clenshaw Curtis rule;

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

SPARSE_GRID_GL, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Gauss-Legendre rule;

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

SPARSE_GRID_HERMITE, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Gauss-Hermite rule;

SPARSE_GRID_MIXED, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules.

SPARSE_GRID_MIXED_GROWTH, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules and a choice of exponential or linear growth rates.

SPARSE_GRID_NCC, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Newton Cotes Closed rule;

SPARSE_GRID_NCO, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Newton Cotes Open rule;

Sample Files:

SGMGA_D2_L1_CCXCC_ANISO is a sparse grid in 2D, at level 1, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D2_L2_CCXCC_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D2_L2_CCXCC_ISO is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with isotropic weights.

SGMGA_D2_L2_CCXGL_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Legendre] with anisotropic weights.

SGMGA_D2_L2_CCXGLG_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Generalized Gauss Laguerre (alpha = 1.5)], with anisotropic weights.

SGMGA_D2_L2_CCXLG_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Laguerre] with anisotropic weights.

SGMGA_D2_L2_F2XGJ_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Fejer Type 2, Gauss Jacobi (alpha = 0.5, beta = 1.5) ] with anisotropic weights.

SGMGA_D2_L2_GGHXGL_ANISO is a sparse grid in 2D, at level 2, whose 1D factors are [Generalized Gauss Hermite ( alpha = 2 ), Gauss Legendre] with anisotropic weights.

SGMGA_D2_L3_CCXCC_ANISO is a sparse grid in 2D, at level 3, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D2_L3_CCXGP_ANISO is a sparse grid in 2D, at level 3, whose 1D factors are [Clenshaw Curtis, Gauss Patterson] with anisotropic weights.

SGMGA_D2_L4_CCXCC_ANISO is a sparse grid in 2D, at level 4, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D2_L5_CCXCC_ANISO is a sparse grid in 2D, at level 5, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D3_L2_CCXCCXCC_ANISO is a sparse grid in 3D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis, Clenshaw Curtis], with anisotropic weights.

SGMGA_D3_L2_CCXCCXCC_ISO is a sparse grid in 3D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis, Clenshaw Curtis], with isotropic weights.

SGMGA_D3_L2_CCXGLXGH_ANISO is a sparse grid in 3D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Legendre, Gauss Hermite] with anisotropic weights.

You can go up one level to the DATASETS page.

Last revised on 14 September 2009.