1Blurb:: Multifidelity uncertainty quantification using stochastic collocation
2
3Description::
4As described in \ref method-stoch_collocation, stochastic collocation
5is a general framework for approximate representation of random response
6functions in terms of finite-dimensional interpolation bases, using
7interpolation polynomials that may be either local or global
8and either value-based or gradient-enhanced.
9
10In the multifidelity case, we decompose this interpolant expansion
11into several constituent expansions, one per model form or solution
12control level.  In a bi-fidelity case with low-fidelity (LF) and
13high-fidelity (HF) models and an additive discrepancy approach, we have:
14
15\f[R = \sum_{i=1}^{N_p^{LF}} r^{LF}_i L_i(\xi)
16     + \sum_{i=1}^{N_p^{HF}} \delta_i L_i(\xi) \f]
17
18where \f$\delta_i\f$ is a coefficient for the discrepancy expansion.
19
20The same specification options are available as described in
21\ref method-stoch_collocation with one key difference: the
22coefficient estimation inputs change from a scalar input for a single
23expansion to a <i>sequence</i> specification for a low-fidelity expansion
24followed by multiple discrepancy expansions.
25
26To obtain the coefficients \f$r_i\f$ and \f$\delta_i\f$ for each of
27the expansions, the following options are provided:
28
29<ol>
30<li> multidimensional integration by a tensor-product of Gaussian
31     quadrature rules (specified with \c quadrature_order_sequence, and,
32     optionally, \c dimension_preference).
33<li> multidimensional integration by the Smolyak sparse grid method
34     (specified with \c sparse_grid_level_sequence and, optionally,
35     \c dimension_preference)
36</ol>
37
38It is important to note that, while \c quadrature_order_sequence and \c
39sparse_grid_level_sequence are
40array inputs, only one scalar from these arrays is active at a time
41for a particular expansion estimation.  In order to specify anisotropy
42in resolution across the random variable set, a \c dimension_preference
43specification can be used to augment these scalar specifications.
44
45Multifidelity UQ using SC requires that the model selected for
46iteration by the method specification is a multifidelity surrogate
47model (see \ref model-surrogate-hierarchical), which defines an
48\c ordered_model_sequence (see \ref model-surrogate-hierarchical).
49Two types of hierarchies are supported: (i) a hierarchy of model forms
50composed from more than one model within the \c ordered_model_sequence,
51or (ii) a hierarchy of discretization levels comprised from a single
52model within the \c ordered_model_sequence which in turn specifies a
53\c solution_level_control (see
54\ref model-single-solution_level_cost-solution_level_control).
55
56In both cases, an expansion will first be formed for the low fidelity
57model or coarse discretization, using the first value within the
58coefficient estimation sequence, along with any specified refinement
59strategy.  Second, expansions are formed for one or more model
60discrepancies (the difference between response results if \c additive
61\c correction or the ratio of results if \c multiplicative \c
62correction), using all subsequent values in the coefficient estimation
63sequence (if the sequence does not provide a new value, then the
64previous value is reused) along with any specified refinement
65strategy.  The number of discrepancy expansions is determined by the
66number of model forms or discretization levels in the hierarchy.
67
68After formation and refinement of the constituent expansions, each of
69the expansions is combined (added or multiplied) into an expansion
70that approximates the high fidelity model, from which the final set of
71statistics are generated.
72
73<b> Additional Resources </b>
74
75%Dakota provides access to multifidelity SC methods through the
76NonDMultilevelStochCollocation class. Refer to the Stochastic Expansion
77Methods chapter of the Theory Manual \cite TheoMan for additional
78information on the Multifidelity SC algorithm.
79
80<b> Expected HDF5 Output </b>
81
82If Dakota was built with HDF5 support and run with the
83\ref environment-results_output-hdf5 keyword, this method
84writes the following results to HDF5:
85
86- \ref hdf5_results-se_moments (expansion moments only)
87- \ref hdf5_results-pdf
88- \ref hdf5_results-level_mappings
89
90In addition, the execution group has the attribute \c equiv_hf_evals, which
91records the equivalent number of high-fidelity evaluations.
92
93Topics::
94
95Examples::
96\verbatim
97method,
98	multifidelity_stoch_collocation
99	  model_pointer = 'HIERARCH'
100	  sparse_grid_level_sequence = 4 3 2
101
102model,
103	id_model = 'HIERARCH'
104	surrogate hierarchical
105	  ordered_model_fidelities = 'LF' 'MF' 'HF'
106	  correction additive zeroth_order
107\endverbatim
108
109Theory::
110
111Faq::
112See_Also::	method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-importance_sampling, method-stoch_collocation
113