Sandia National Laboratories

# greedy

Sample allocation based on greedy refinement within multifidelity stochastic collocation

Specification

- Alias: None

- Arguments: None

Description

Multifidelity stochastic collocation supports greedy refinement strategies using tensor and sparse grids for both nodal and hierarchical collocation approaches. The key idea is that each level of the model hierarchy being approximated can generate one or more candidates for refinement. These candidates are competed against each other within an integrated competition, and the candidate that induces the largest change in the statistical QoI (response covariance by default, or results of any level mappings when specified), normalized by relative cost of evaluating the candidate, is selected and then used to generate additional candidates for consideration at its model level.

Examples

The following example of greedy multifidelity stochastic collocation using nodel interpolation starts from a zeroth-order expansion (a constant) for each level, and generates uniform candidate refinements for each level that are competed in a greedy competition. The number of new samples for the incremented candidate expansion order is determined from the quadrature rules of the new sparse grid level. In this case, the number of candidates for each level is limited to one uniform refinement of the current sparse grid level.

    method,
     model_pointer = 'HIERARCH'
     multifidelity_stoch_collocation
       nodal
       allocation_control greedy
       p_refinement uniform
         sparse_grid_level_sequence = 0 unrestricted
         convergence_tolerance 1.e-3

The next example employs generalized sparse grids and hierarchical interpolation. Each level starts from a level 0 reference grid (a single point) and generates multiple admissible index set candidates. The full set of candidates across all levels is competed within a unified greedy competition, where the greedy selection metric is the induced change in the statistical QoI, normalized by the aggregate simulation cost of the index set candidate. In this case, there are multiple candidates for each level and the number of candidates grows rapidly with random dimension and grid level.

    method,
     model_pointer = 'HIERARCH'
     multifidelity_stoch_collocation
       hierarchical
       allocation_control greedy
       p_refinement dimension_adaptive generalized
         sparse_grid_level_sequence = 0 unrestricted
         convergence_tolerance 1.e-8

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