1Blurb:: Multilevel uncertainty quantification using polynomial chaos expansions 2 3Description:: 4As described in \ref method-polynomial_chaos, the polynomial chaos 5expansion (PCE) is a general framework for the approximate representation 6of random response functions in terms of series expansions in standardized 7random variables: 8 9\f[R = \sum_{i=0}^P \alpha_i \Psi_i(\xi) \f] 10 11where \f$\alpha_i\f$ is a deterministic coefficient, \f$\Psi_i\f$ is a 12multidimensional orthogonal polynomial and \f$\xi\f$ is a vector of 13standardized random variables. 14 15In the multilevel and multifidelity cases, we decompose this expansion 16into several constituent expansions, one per model form or solution 17control level. In a bi-fidelity case with low-fidelity (LF) and 18high-fidelity (HF) models, we have: 19 20\f[R = \sum_{i=0}^{P^{LF}} \alpha^{LF}_i \Psi_i(\xi) 21 + \sum_{i=0}^{P^{HF}} \delta_i \Psi_i(\xi) \f] 22 23where \f$\delta_i\f$ is a coefficient for the discrepancy expansion. 24 25 26 27For the case of regression-based PCE (least squares, compressed sensing, 28or orthogonal least interpolation), an optimal sample allocation procedure 29can be applied for the resolution of each level within a multilevel sampling 30procedure as in \ref method-multilevel_sampling. The core difference 31is that a Monte Carlo estimator of the statistics is replaced with a 32PCE-based estimator of the statistics, requiring approximation of the 33variance of these estimators. 34 35Initial prototypes for multilevel PCE can be explored using \c 36dakota/share/dakota/test/dakota_uq_diffusion_mlpce.in, and will be stabilized in 37future releases. 38 39 40<b> Additional Resources </b> 41 42%Dakota provides access to multilevel PCE methods through the 43NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion 44Methods chapter of the Theory Manual \cite TheoMan for additional 45information on the Multilevel PCE algorithm. 46 47<b> Expected HDF5 Output </b> 48 49If Dakota was built with HDF5 support and run with the 50\ref environment-results_output-hdf5 keyword, this method 51writes the following results to HDF5: 52 53- \ref hdf5_results-se_moments (expansion moments only) 54- \ref hdf5_results-pdf 55- \ref hdf5_results-level_mappings 56 57In addition, the execution group has the attribute \c equiv_hf_evals, which 58records the equivalent number of high-fidelity evaluations. 59 60Topics:: 61 62Examples:: 63\verbatim 64method, 65 multilevel_polynomial_chaos 66 model_pointer = 'HIERARCH' 67 pilot_samples = 10 68 expansion_order_sequence = 2 69 collocation_ratio = .9 70 seed = 1237 71 orthogonal_matching_pursuit 72 convergence_tolerance = .01 73 output silent 74 75model, 76 id_model = 'HIERARCH' 77 surrogate hierarchical 78 ordered_model_fidelities = 'SIM1' 79 correction additive zeroth_order 80 81model, 82 id_model = 'SIM1' 83 simulation 84 solution_level_control = 'mesh_size' 85 solution_level_cost = 1. 8. 64. 512. 4096. 86\endverbatim 87 88Theory:: 89 90Faq:: 91See_Also:: method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-importance_sampling, method-stoch_collocation 92