Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Cubature using tensorproducts of Gaussian quadrature rules
Alias: none
Argument(s): INTEGERLIST
Multidimensional integration by a tensorproduct of Gaussian quadrature rules (specified with quadrature_order
, and, optionally, dimension_preference
). The default rule selection is to employ non_nested
Gauss rules including GaussHermite (for normals or transformed normals), GaussLegendre (for uniforms or transformed uniforms), GaussJacobi (for betas), GaussLaguerre (for exponentials), generalized GaussLaguerre (for gammas), and numericallygenerated Gauss rules (for other distributions when using an Extended basis). For the case of p_refinement
or the case of an explicit nested
override, GaussHermite rules are replaced with GenzKeister nested rules and GaussLegendre rules are replaced with GaussPatterson nested rules, both of which exchange lower integrand precision for greater point reuse. By specifying a dimension_preference
, where higher preference leads to higher order polynomial resolution, the tensor grid may be rendered anisotropic. The dimension specified to have highest preference will be set to the specified quadrature_order
and all other dimensions will be reduced in proportion to their reduced preference; any nonintegral portion is truncated. To synchronize with tensorproduct integration, a tensorproduct expansion is used, where the order of the expansion in each dimension is selected to be half of the integrand precision available from the rule in use, rounded down. In the case of nonnested Gauss rules with integrand precision , is one less than the quadrature order in each dimension (a onedimensional expansion contains the same number of terms, , as the number of Gauss points). The total number of terms, N, in a tensorproduct expansion involving n uncertain input variables is
In some advanced use cases (e.g., multifidelity UQ), multiple grid resolutions can be employed; for this reason, the quadrature_order
specification supports an array input.