sandia_sgmga, a C++ 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.
sandia_sgmga is a variant of the SGMGA library. It has the same capabilities as that library, but it uses a different interface to the SANDIA_RULES routines which compute points and weights of quadrater rules. Instead of passing the parameters needed by some of those rules as function arguments, these values are made available indirectly. This library implements that indirect storage using a function called parameter, which must therefore be supplied by the user as part of every program that calls the library. Refer to the source code of the example programs to see how parameter is defined and used.
sandia_sgmga calls routines from the SANDIA_RULES and SANDIA_RULES2 libraries. Source code or compiled copies of those libraries must be available when a program wishes to use the sandia_sgmga library.
| Name | Abbreviation | Interval | Weight function |
|---|---|---|---|
| Clenshaw-Curtis | CC | [-1,+1] | 1 |
| Fejer Type 2 | F2 | [-1,+1] | 1 |
| Gauss Patterson | GP | [-1,+1] | 1 |
| Gauss-Legendre | GL | [-1,+1] | 1 |
| Gauss-Hermite | GH | (-oo,+oo) | e-x*x |
| Generalized Gauss-Hermite | GGH | (-oo,+oo) | |x|alpha e-x*x |
| Gauss-Laguerre | LG | [0,+oo) | e-x |
| Generalized Gauss-Laguerre | GLG | [0,+oo) | xalpha e-x |
| Gauss-Jacobi | GJ | [-1,+1] | (1-x)alpha (1+x)beta |
| Hermite Genz-Keister | HGK | (-oo,+oo) | e-x*x |
In the sparse grid setting, for any 1D quadrature rule, it is necessary to select a sequence of rules of increasing order (number of points), indexed by a variable we will call the "level". Thus, although the Clenshaw Curtis rule can be set up for any, a common procedure in sparse grids is to choose select the rules of order 1, 3, 5, 9, 17, 33, ..., assigning these the levels 0, 1, 2, 3, 4, 5 and so forth. The relationship between level (L) and order (O) will be called the growth rule.
The details of growth rules vary somewhat, depending on whether there is nesting to take advantage of, whether the user wants to economize as much as possible in the number of points added, and so on. For each dimension, the user must specify a growth rule appropriate for the chosen quadrature rule. We provide a number of predefined growth rules that are suitable.
Here are the names of the growth rule functions, with a brief comment on their behavior and use. These growth rule functions are available in the sandia_rules library where their details may be examined.
| Growth Rule | Discussion |
|---|---|
| level_to_order_exp_cc() | Clenshaw Curtis rule. Fast growth is exponential |
| level_to_order_exp_f2() | Fejer Type 2 rule. Fast growth is exponential |
| level_to_order_exp_gauss() | Gaussian rules. Fast growth is exponential, O=2^(L+1)-1 |
| level_to_order_exp_hgk() | Genz-Keister rules for Hermite weight; |
| level_to_order_linear_nn() | Linear growth for a non-nested rule; |
| level_to_order_linear_wn() | Linear growth for a weakly-nested rule (typically, an abscissas at 0 is common); |
Each growth rule has "slow", "moderate" and "fast" settings. A scalar quantity GROWTH selects the rule order O for level L from the three growth options for each 1D rule. In the case of exponentially growing rules, the slow and moderate growth rules choose O indirectly, by imposing a requirement on P, the degree of precision of the rule.
| Value | Name | Meaning |
|---|---|---|
| 0 | Slow | O=L+1 for linear rules, P=2*L+1 for exponential |
| 1 | Moderate | O=2*L+1 for linear rules, P=4*L+1 for exponential |
| 2 | Full | O=2*L+1 for linear rules, O = next rule in sequence for exponential |
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
sandia_sgmga is available in a C++ version.
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