SGMG is a dataset directory which contains some examples of sparse grids in which different 1D quadrature rules may be used for each spatial dimension.
These sparse grids are different from those stored in the SPARSE_GRID_MIXED dataset directory. These sparse grids have a variation in the way the 1D rules grow with the level. Essentially, the original rule was "exponential" in every case. Roughly speaking, the next 1D level used twice as many points.
In the SGMG family, however, the default growth rate depends on the rule. Thus:
The following list includes 1D quadrature rules that may be specified as factors. A few rules are not yet supported.
Each sparse grid is stored using the "weighted quadrature rule" format, that is, as 5 files:
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.
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
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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.
SGMG, a C++ library which constructs a multidimensional sparse grid from mixed 1D factors and with a choice of exponential and linear growth rates.
SGMGA, a dataset directory which contains SGMGA files (Sparse Grid Mixed Growth Anisotropic), that is, multidimensional Smolyak sparse grids based on a mixture of 1D rules, and with a choice of exponential and linear growth rates for the 1D rules and anisotropic weights for the dimensions.
SPARSE_GRID_CC, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Clenshaw Curtis rule;
SPARSE_GRID_DISPLAY, a MATLAB program which reads a file of points on a sparse grid, displays the grid and saves the image in a PNG file;
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_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;
SPARSE_GRIDS_DISPLAY, a MATLAB program which reads two files of sparse grids, displays the first with hollow blue dots, the second with red dots.
SGMG_D2_L2_CCXCC is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis]
SGMG_D2_L2_CCXGL is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Legendre]
SGMG_D2_L2_CCXGLG is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Generalized Gauss Laguerre (alpha = 1.5)].
SGMG_D2_L2_CCXLG is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Laguerre]
SGMG_D2_L2_F2XGJ is a sparse grid in 2D, at level 2, whose 1D factors are [Fejer Type 2, Gauss Jacobi (alpha = 0.5, beta = 1.5) ]
SGMG_D2_L2_GGHXGL is a sparse grid in 2D, at level 2, whose 1D factors are [Generalized Gauss Hermite ( alpha = 2 ), Gauss Legendre]
SGMG_D2_L3_CCXGP is a sparse grid in 2D, at level 3, whose 1D factors are [Clenshaw Curtis, Gauss Patterson]
SGMG_D3_L2_CCXF2XGH is a sparse grid in 3D, at level 2, whose 1D factors are [Clenshaw Curtis, Fejer Type 2, Gauss Hermite]
You can go up one level to the DATASETS page.