Blurb:: Multilevel uncertainty quantification using polynomial chaos expansions

Description:: 
As described in \ref method-polynomial_chaos, the polynomial chaos 
expansion (PCE) is a general framework for the approximate representation 
of random response functions in terms of series expansions in standardized 
random variables:

\f[R = \sum_{i=0}^P \alpha_i \Psi_i(\xi) \f]

where \f$\alpha_i\f$ is a deterministic coefficient, \f$\Psi_i\f$ is a
multidimensional orthogonal polynomial and \f$\xi\f$ is a vector of
standardized random variables. 

In the multilevel and multifidelity cases, we decompose this 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, we have:

\f[R = \sum_{i=0}^{P^{LF}} \alpha^{LF}_i \Psi_i(\xi)
     + \sum_{i=0}^{P^{HF}}      \delta_i \Psi_i(\xi) \f]

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



For the case of regression-based PCE (least squares, compressed sensing, 
or orthogonal least interpolation), an optimal sample allocation procedure 
can be applied for the resolution of each level within a multilevel sampling
procedure as in \ref method-multilevel_sampling.  The core difference
is that a Monte Carlo estimator of the statistics is replaced with a
PCE-based estimator of the statistics, requiring approximation of the 
variance of these estimators.

Initial prototypes for multilevel PCE can be explored using \c
dakota/share/dakota/test/dakota_uq_diffusion_mlpce.in, and will be stabilized in
future releases.


<b> Additional Resources </b>

%Dakota provides access to multilevel PCE methods through the
NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion
Methods chapter of the Theory Manual \cite TheoMan for additional
information on the Multilevel PCE 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,
	multilevel_polynomial_chaos
	  model_pointer = 'HIERARCH'
	  pilot_samples = 10
	  expansion_order_sequence = 2
	  collocation_ratio = .9
	  seed = 1237
	  orthogonal_matching_pursuit
	  convergence_tolerance = .01
	output silent

model,
	id_model = 'HIERARCH'
	surrogate hierarchical
	  ordered_model_fidelities = 'SIM1'
	  correction additive zeroth_order

model,
	id_model = 'SIM1'
	simulation
	  solution_level_control = 'mesh_size'
	  solution_level_cost = 1. 8. 64. 512. 4096.
\endverbatim

Theory::

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