Blurb::
Multifidelity uncertainty quantification using function train expansions
Description::
As described in the :dakkw:`method-function_train` method and the
:dakkw:`model-surrogate-global-function_train` model,
the function train (FT) approximation is a polynomial expansion that exploits low rank
structure within the mapping from input random variables to output quantities of interest
(QoI).  For multilevel and multifidelity function train approximations, we decompose this
expansion into several constituent expansions, one per model form or solution control
level, where independent function train approximations are constructed for the
low-fidelity/coarse resolution model and one or more levels of model discrepancy.

In a three-model case with low-fidelity (L), medium-fidelity (M), and
high-fidelity (H) models and an additive discrepancy approach, we can denote this as:


.. math::  Q^H \approx \hat{Q}_{r_L}^L + \hat{\Delta}_{r_{ML}}^{ML} + \hat{\Delta}_{r_{HM}}^{HM} 

where :math:`\Delta^{ij}`  represents a discrepancy expansion computed from
:math:`Q^i - Q^j`  and reduced rank representations of these discrepancies may
be targeted ( :math:`r_{HM} < r_{ML} < r_L` ).

In multifidelity approaches, sample allocation for the constituent expansions can be
performed with either no, individual, or integrated adaptive refinement as described in
:dakkw:`method-multifidelity_function_train-allocation_control`.

*Expected HDF5 Output*

If Dakota was built with HDF5 support and run with the
:dakkw:`environment-results_output-hdf5` keyword, this method
writes the following results to HDF5:


- :ref:`hdf5_results-se_moments` (expansion moments only)
- :ref:`hdf5_results-pdf`
- :ref:`hdf5_results-level_mappings`

In addition, the execution group has the attribute ``equiv_hf_evals``, which
records the equivalent number of high-fidelity evaluations.
Topics::

Examples::

Theory::

Faq::

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