Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Automatic polynomial order refinement
Alias: none
Argument(s): none
Default: no refinement
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Required (Choose One)  prefinement type (Group 1)  uniform  Refine an expansion uniformly in all dimensions.  
dimension_adaptive  Perform anisotropic expansion refinement by preferentially adapting in dimensions that are detected to have higher `importance'. 
The p_refinement
keyword specifies the usage of automated polynomial order refinement, which can be either uniform
or dimension_adaptive
.
The dimension_adaptive
option is supported for the tensorproduct quadrature and Smolyak sparse grid options and uniform
is supported for tensor and sparse grids as well as regression approaches (collocation_points
or collocation_ratio
).
Each of these refinement cases makes use of the max_iterations
and convergence_tolerance
method independent controls. The former control limits the number of refinement iterations, and the latter control terminates refinement when the twonorm of the change in the response covariance matrix (or, in goaloriented approaches, the twonorm of change in the statistical quantities of interest (QOI)) falls below the tolerance.
The dimension_adaptive
case can be further specified to utilize sobol
, decay
, or generalized
refinement controls. The former two cases employ anisotropic tensor/sparse grids in which the anisotropic dimension preference (leading to anisotropic integrations/expansions with differing refinement levels for different random dimensions) is determined using either total Sobol' indices from variancebased decomposition (sobol
case: high indices result in high dimension preference) or using spectral coefficient decay rates from a rate estimation technique similar to Richardson extrapolation (decay
case: low decay rates result in high dimension preference). In these two cases as well as the uniform
refinement case, the quadrature_order
or sparse_grid_level
are ramped by one on each refinement iteration until either of the two convergence controls is satisfied. For the uniform
refinement case with regression approaches, the expansion_order
is ramped by one on each iteration while the oversampling ratio (either defined by collocation_ratio
or inferred from collocation_points
based on the initial expansion) is held fixed. Finally, the generalized
dimension_adaptive
case is the default adaptive approach; it refers to the generalized sparse grid algorithm, a greedy approach in which candidate index sets are evaluated for their impact on the statistical QOI, the most influential sets are selected and used to generate additional candidates, and the index set frontier of a sparse grid is evolved in an unstructured and goaloriented manner (refer to User's Manual PCE descriptions for additional specifics).
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.