Blurb:: Multifidelity uncertainty quantification using stochastic collocation

Description:: 
As described in \ref method-stoch_collocation, stochastic collocation 
is a general framework for approximate representation of random response 
functions in terms of finite-dimensional interpolation bases, using
interpolation polynomials that may be either local or global
and either value-based or gradient-enhanced.

In the multifidelity case, we decompose this interpolant expansion
into several constituent expansions, one per model form or solution
control level.  In a bi-fidelity case with low-fidelity (LF) and
high-fidelity (HF) models and an additive discrepancy approach, we have:

\f[R = \sum_{i=1}^{N_p^{LF}} r^{LF}_i L_i(\xi)
     + \sum_{i=1}^{N_p^{HF}} \delta_i L_i(\xi) \f]

where \f$\delta_i\f$ is a coefficient for the discrepancy expansion.

The same specification options are available as described in 
\ref method-stoch_collocation with one key difference: the
coefficient estimation inputs change from a scalar input for a single
expansion to a <i>sequence</i> specification for a low-fidelity expansion
followed by multiple discrepancy expansions.

To obtain the coefficients \f$r_i\f$ and \f$\delta_i\f$ for each of 
the expansions, the following options are provided:

<ol>
<li> multidimensional integration by a tensor-product of Gaussian
     quadrature rules (specified with \c quadrature_order_sequence, and,
     optionally, \c dimension_preference).
<li> multidimensional integration by the Smolyak sparse grid method
     (specified with \c sparse_grid_level_sequence and, optionally, 
     \c dimension_preference)
</ol>

It is important to note that, while \c quadrature_order_sequence and \c
sparse_grid_level_sequence are
array inputs, only one scalar from these arrays is active at a time
for a particular expansion estimation.  In order to specify anisotropy
in resolution across the random variable set, a \c dimension_preference 
specification can be used to augment these scalar specifications.

Multifidelity UQ using SC requires that the model selected for
iteration by the method specification is a multifidelity surrogate
model (see \ref model-surrogate-hierarchical), which defines an
\c ordered_model_sequence (see \ref model-surrogate-hierarchical).
Two types of hierarchies are supported: (i) a hierarchy of model forms
composed from more than one model within the \c ordered_model_sequence,
or (ii) a hierarchy of discretization levels comprised from a single
model within the \c ordered_model_sequence which in turn specifies a 
\c solution_level_control (see 
\ref model-single-solution_level_cost-solution_level_control).

In both cases, an expansion will first be formed for the low fidelity
model or coarse discretization, using the first value within the
coefficient estimation sequence, along with any specified refinement
strategy.  Second, expansions are formed for one or more model
discrepancies (the difference between response results if \c additive
\c correction or the ratio of results if \c multiplicative \c
correction), using all subsequent values in the coefficient estimation
sequence (if the sequence does not provide a new value, then the
previous value is reused) along with any specified refinement
strategy.  The number of discrepancy expansions is determined by the
number of model forms or discretization levels in the hierarchy.

After formation and refinement of the constituent expansions, each of
the expansions is combined (added or multiplied) into an expansion
that approximates the high fidelity model, from which the final set of
statistics are generated.

<b> Additional Resources </b>

%Dakota provides access to multifidelity SC methods through the
NonDMultilevelStochCollocation class. Refer to the Stochastic Expansion
Methods chapter of the Theory Manual \cite TheoMan for additional
information on the Multifidelity SC algorithm.

<b> Expected HDF5 Output </b>

If Dakota was built with HDF5 support and run with the 
\ref 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 \c equiv_hf_evals, which
records the equivalent number of high-fidelity evaluations.

Topics::

Examples::
\verbatim
method,
	multifidelity_stoch_collocation
	  model_pointer = 'HIERARCH'
	  sparse_grid_level_sequence = 4 3 2

model,
	id_model = 'HIERARCH'
	surrogate hierarchical
	  ordered_model_fidelities = 'LF' 'MF' 'HF'
	  correction additive zeroth_order
\endverbatim

Theory::

Faq::
See_Also::	method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-importance_sampling, method-stoch_collocation
