1Blurb:: Multilevel uncertainty quantification using function train expansions
2
3Description::
4
5As described in the \ref method-function_train method and the
6\ref model-surrogate-global-function_train model,
7the function train (FT) approximation is a polynomial expansion that exploits low rank
8structure within the mapping from input random variables to output quantities of interest
9(QoI).  For multilevel and multifidelity function train approximations, we decompose this
10expansion into several constituent expansions, one per model form or solution control
11level, where independent function train approximations are constructed for the
12low-fidelity/coarse resolution model and one or more levels of model discrepancy.
13
14In a three-model case with low-fidelity (L), medium-fidelity (M), and
15high-fidelity (H) models and an additive discrepancy approach, we can denote this as:
16
17\f[ Q^H \approx \hat{Q}_{r_L}^L + \hat{\Delta}_{r_{ML}}^{ML} + \hat{\Delta}_{r_{HM}}^{HM} \f]
18
19where \f$\Delta^{ij}\f$ represents a discrepancy expansion computed from
20\f$Q^i - Q^j\f$ and reduced rank representations of these discrepancies may
21be targeted (\f$ r_{HM} < r_{ML} < r_L \f$).
22
23In multilevel approaches, sample allocation for the constituent expansions is
24performed as described in \ref method-multilevel_function_train-allocation_control.
25
26<b> Expected HDF5 Output </b>
27
28If Dakota was built with HDF5 support and run with the
29\ref environment-results_output-hdf5 keyword, this method
30writes the following results to HDF5:
31
32- \ref hdf5_results-se_moments (expansion moments only)
33- \ref hdf5_results-pdf
34- \ref hdf5_results-level_mappings
35
36In addition, the execution group has the attribute \c equiv_hf_evals, which
37records the equivalent number of high-fidelity evaluations.
38
39Topics::
40
41Examples::
42Theory::
43Faq::
44See_Also:: model-surrogate-global-function_train, method-function_train
45