sparse_grid_level

Level to use in sparse grid integration or interpolation

Specification

• Alias: None

• Arguments: INTEGER

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		dimension_preference	A set of weights specifying the realtive importance of each uncertain variable (dimension)
Optional (Choose One)	Optional (Choose One)	nodal	Employ a nodal sparse grid construction in stochastic collocation
		hierarchical	Employ a hierarchical sparse grid construction
Optional (Choose One)	Quadrature Rule Growth	restricted	Restrict the growth rates for nested and non-nested rules can be synchronized for consistency.
		unrestricted	Overide the default restriction of growth rates for nested and non-nested rules that are by defualt synchronized for consistency.
Optional (Choose One)	Quadrature Rule Nesting	nested	Enforce use of nested quadrature rules if available
		non_nested	Enforce use of non-nested quadrature rules

Description

Multi-dimensional integration by the Smolyak sparse grid method (specified with sparse_grid_level and, optionally, dimension_preference). The underlying one-dimensional integration rules are the same as for the tensor-product quadrature case; however, the default rule selection is nested for sparse grids (Genz-Keister for normals/transformed normals and Gauss-Patterson for uniforms/transformed uniforms). This default can be overridden with an explicit non_nested specification (resulting in Gauss-Hermite for normals/transformed normals and Gauss-Legendre for uniforms/transformed uniforms). As for tensor quadrature, the dimension_preference specification enables the use of anisotropic sparse grids (refer to the PCE description in the User’s Manual for the anisotropic index set constraint definition). Similar to anisotropic tensor grids, the dimension with greatest preference will have resolution at the full sparse_grid_level and all other dimension resolutions will be reduced in proportion to their reduced preference. For PCE with either isotropic or anisotropic sparse grids, a summation of tensor-product expansions is used, where each anisotropic tensor-product quadrature rule underlying the sparse grid construction results in its own anisotropic tensor-product expansion as described in case 1. These anisotropic tensor-product expansions are summed into a sparse PCE using the standard Smolyak summation (again, refer to the User’s Manual for additional details). As for quadrature_order, the sparse_grid_level specification admits an array input for enabling specification of multiple grid resolutions used by certain advanced solution methodologies.

A corresponding sequence specification is documented at, e.g., sparse_grid_level_sequence and sparse_grid_level_sequence