Automatic polynomial order refinement


  • Alias: None

  • Arguments: None

  • Default: no refinement

Child Keywords:


Description of Group

Dakota Keyword

Dakota Keyword Description

Required (Choose One)

p-refinement Type


Refine an expansion uniformly in all dimensions.


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 tensor-product 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 two-norm of the change in the response covariance matrix (or, in goal-oriented approaches, the two-norm 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 variance-based 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 goal-oriented 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, Gauss-Hermite rules are replaced with Genz-Keister nested rules and Gauss-Legendre rules are replaced with Gauss-Patterson nested rules, both of which exchange lower integrand precision for greater point reuse.