sgmga, a Fortran90 code which implements a family of sparse grid rules. These rules are mixed, in that a different 1D quadrature rule can be specified for each dimension. Moreover, each 1D quadrature rule comes in a family of increasing size whose growth rate (typically linear or exponential) is chosen by the user. Finally, the user may also specify different weights for each dimension, resulting in anisotropic rules.
sgmga() calls many routines from the SANDIA_RULES() code. Source code or compiled copies of both libraries must be available when a program wishes to use sgmga().
Thanks to Drew Kouri, who pointed out a discrepancy in the computation of the variable level_1d_max which meant that certain sparse grids requested the generation of a 1D rule of a level that was higher than necessary by 1. In particular, if the Gauss-Patterson rule was involved, sparse grids that actually only needed rules of level 7 would ask also for level 8, resulting in the computation being terminated. This problem was corrected on 25 April 2011.
| Index | Name | Abbreviation | Default Growth Rule | Interval | Weight function |
|---|---|---|---|---|---|
| 1 | Clenshaw-Curtis | CC | Moderate Exponential | [-1,+1] | 1 |
| 2 | Fejer Type 2 | F2 | Moderate Exponential | [-1,+1] | 1 |
| 3 | Gauss Patterson | GP | Moderate Exponential | [-1,+1] | 1 |
| 4 | Gauss-Legendre | GL | Moderate Linear | [-1,+1] | 1 |
| 5 | Gauss-Hermite | GH | Moderate Linear | (-oo,+oo) | e-x*x |
| 6 | Generalized Gauss-Hermite | GGH | Moderate Linear | (-oo,+oo) | |x|alpha e-x*x |
| 7 | Gauss-Laguerre | LG | Moderate Linear | [0,+oo) | e-x |
| 8 | Generalized Gauss-Laguerre | GLG | Moderate Linear | [0,+oo) | xalpha e-x |
| 9 | Gauss-Jacobi | GJ | Moderate Linear | [-1,+1] | (1-x)alpha (1+x)beta |
| 10 | Hermite Genz-Keister | HGK | Moderate Exponential | (-oo,+oo) | e-x*x |
| 11 | User Supplied Open | UO | Moderate Linear | ? | ? |
| 12 | User Supplied Closed | UC | Moderate Linear | ? | ? |
For a given family, a growth rule can be prescribed, which determines the orders O of the sequence of rules selected from the family. The selected rules are indexed by L, which starts at 0. The polynomial precision P of the rule is sometimes used to determine the appropriate order O.
| Index | Name | Order Formula |
|---|---|---|
| 0 | Default | "DF", moderate exponential or moderate linear |
| 1 | "SL", Slow linear | O=L+1 |
| 2 | "SO", Slow Linear Odd | O=1+2*((L+1)/2) |
| 3 | "ML", Moderate Linear | O=2L+1 |
| 4 | "SE", Slow Exponential | select smallest exponential order O so that 2L+1 <= P |
| 5 | "ME", Moderate Exponential | select smallest exponential order O so that 4L+1 <= P |
| 6 | "FE", Full Exponential | O=2^L+1 for Clenshaw Curtis, O=2^(L+1)-1 otherwise. |
A version of the sparse grid code is available in https://tasmanian.ornl.gov, the TASMANIAN code, available from Oak Ridge National Laboratory.
The information on this web page is distributed under the MIT license.
sgmga is available in a C version and a C++ version and a Fortran90 version and a MATLAB version.
nint_exactness_mixed, a Fortran90 code which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.
quad_rule, a Fortran90 code which defines quadrature rules for various intervals and weight functions.
sandia_rules, a Fortran90 code which generates Gauss quadrature rules of various orders and types.
sandia_sparse, a Fortran90 code which computes the points and weights of a Smolyak sparse grid, based on a variety of 1-dimensional quadrature rules.
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.
smolpack, a C code which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids, by Knut Petras.
sparse_grid_mixed, a Fortran90 code which creates a sparse grid dataset based on a mixed set of 1D factor rules.
toms847, a MATLAB code which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as spinterp(), and is by Andreas Klimke.