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