p_refinement
Automatic polynomial order refinement
Specification
Alias: None
Arguments: None
Default: no refinement
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Required (Choose One) 
prefinement Type 
Refine an expansion uniformly in all dimensions. 

Perform anisotropic expansion refinement by preferentially adapting in dimensions that are detected to have higher “importance”. 
Description
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.