1Blurb:: Sample allocation based on greedy refinement within multifidelity polynomial chaos 2 3Description:: 4Multifidelity polynomial chaos supports greedy 5refinement strategies, spanning regression and projection approaches 6for computing expansion coefficients. The key idea is that each level 7of the model hierarchy being approximated can generate one or more 8candidates for refinement. These candidates are competed against each 9other within a unified competition, and the candidate that induces the 10largest change in the statistical QoI (response covariance by default, 11or results of any \f$z/p/\beta/\beta^*\f$ level mappings when specified), 12normalized by relative cost of evaluating the candidate, is selected 13and then used to generate additional candidates for consideration at 14its model level. 15 16Topics:: 17 18Examples:: 19 20The following example of greedy multifidelity regression starts from a 21zeroth-order reference expansion (a constant) for each level, and 22generates candidate refinements for each level that are competed in an 23integrated greedy competition. The number of new samples for the 24incremented candidate expansion order is determined from the collocation 25ratio. In this case, the number of candidates for each level is limited 26to one uniform refinement of the current expansion order. 27 28\verbatim 29method, 30 model_pointer = 'HIERARCH' 31 multifidelity_polynomial_chaos 32 allocation_control greedy 33 p_refinement uniform 34 expansion_order_sequence = 0 35 collocation_ratio = .9 seed = 160415 36 orthogonal_matching_pursuit 37 convergence_tolerance 1.e-2 38\endverbatim 39 40The next example employs generalized sparse grids within a greedy 41multifidelity competition. Each modeling level starts from a level 0 42reference grid (a single point) and generates multiple admissible 43index set candidates. The full set of candidates across all model 44levels is competed within an integrated greedy competition, where the 45greedy selection metric is the induced change in the statistical QoI, 46normalized by the aggregate simulation cost of the index set 47candidate. In this case, there are multiple candidates for each model 48level and the number of candidates grows rapidly with random dimension 49and grid level. 50 51\verbatim 52method, 53 model_pointer = 'HIERARCH' 54 multifidelity_polynomial_chaos 55 allocation_control greedy 56 p_refinement dimension_adaptive generalized 57 sparse_grid_level_sequence = 0 unrestricted 58 convergence_tolerance 1.e-8 59\endverbatim 60 61 62Theory:: 63 64Faq:: 65See_Also:: 66