SPARSE_GRID_MIXED
Sparse Grids from Mixed 1D Rules

SPARSE_GRID_MIXED 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 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.
  (not supported yet)

Each sparse grid is stored using the "weighted quadrature rule" format, that is, as 5 files:

• an "A" or "alpha" file, which lists the value of ALPHA for each dimension;
• an "B" or "beta" file, which lists the value of BETA for each dimension;
• 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 weights alpha and beta will generally be zero. However, a generalized Laguerre or Hermite rule will have an associated value of alpha, and the Jacobi rule will have values of both alpha and beta.

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
      

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
      

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:

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 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_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 C++ library which constructs a multidimensional sparse grid from mixed 1D factors.

SPARSE_GRID_MIXED_GROWTH, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules and user-specified growth 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;

Sample Files:

SPARSE_GRID_MIXED_D2_L2_CCXCC is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Clenshaw Curtis]

• sparse_grid_mixed_d2_l2_ccxcc_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxcc_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxcc_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxcc_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxcc_x.txt, the X file.

SPARSE_GRID_MIXED_D2_L2_CCXGL is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Legendre]

• sparse_grid_mixed_d2_l2_ccxgl_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxgl_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxgl_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxgl_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxgl_x.txt, the X file.

SPARSE_GRID_MIXED_D2_L2_CCXGLG is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Generalized Gauss Laguerre (alpha = 1.5)].

• sparse_grid_mixed_d2_l2_ccxglg_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxglg_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxglg_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxglg_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxglg_x.txt, the X file.

SPARSE_GRID_MIXED_D2_L2_CCXLG is a sparse grid in 2D, at level 2, whose 1D factors are [Clenshaw Curtis, Gauss Laguerre]

• sparse_grid_mixed_d2_l2_ccxlg_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxlg_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxlg_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxlg_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxlg_x.txt, the X file.

SPARSE_GRID_MIXED_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) ]

• sparse_grid_mixed_d2_l2_f2xgj_a.txt, the A file.
• sparse_grid_mixed_d2_l2_f2xgj_b.txt, the B file.
• sparse_grid_mixed_d2_l2_f2xgj_r.txt, the R file.
• sparse_grid_mixed_d2_l2_f2xgj_w.txt, the W file.
• sparse_grid_mixed_d2_l2_f2xgj_x.txt, the X file.

SPARSE_GRID_MIXED_D2_L2_GGHXGL is a sparse grid in 2D, at level 2, whose 1D factors are [Generalized Gauss Hermite ( alpha = 2 ), Gauss Legendre]

• sparse_grid_mixed_d2_l2_gghxgl_a.txt, the A file.
• sparse_grid_mixed_d2_l2_gghxgl_b.txt, the B file.
• sparse_grid_mixed_d2_l2_gghxgl_r.txt, the R file.
• sparse_grid_mixed_d2_l2_gghxgl_w.txt, the W file.
• sparse_grid_mixed_d2_l2_gghxgl_x.txt, the X file.

SPARSE_GRID_MIXED_D2_L3_CCXGP is a sparse grid in 2D, at level 3, whose 1D factors are [Clenshaw Curtis, Gauss Patterson]

• sparse_grid_mixed_d2_l3_ccxgp_a.txt, the A file.
• sparse_grid_mixed_d2_l3_ccxgp_b.txt, the B file.
• sparse_grid_mixed_d2_l3_ccxgp_r.txt, the R file.
• sparse_grid_mixed_d2_l3_ccxgp_w.txt, the W file.
• sparse_grid_mixed_d2_l3_ccxgp_x.txt, the X file.

SPARSE_GRID_MIXED_D3_L2_CCXF2XGH is a sparse grid in 3D, at level 2, whose 1D factors are [Clenshaw Curtis, Fejer Type 2, Gauss Hermite]

• sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt, the A file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt, the B file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt, the R file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt, the W file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt, the X file.

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