Blurb::
Multilevel uncertainty quantification using polynomial chaos expansions
Description::
As described in :dakkw:`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:


.. math:: R = \sum_{i=0}^P \alpha_i \Psi_i(\xi) 

where :math:`\alpha_i`  is a deterministic coefficient, :math:`\Psi_i`  is a
multidimensional orthogonal polynomial and :math:`\xi`  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:


.. math:: R = \sum_{i=0}^{P^{LF}} \alpha^{LF}_i \Psi_i(\xi)
     + \sum_{i=0}^{P^{HF}}      \delta_i \Psi_i(\xi) 

where :math:`\delta_i`  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 :dakkw:`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
``dakota/share/dakota/test/dakota_uq_diffusion_mlpce``.in, and will be stabilized in
future releases.


*Additional Resources*

Dakota provides access to multilevel PCE methods through the
NonDMultilevelPolynomialChaos class. Refer to the 
:ref:`theory:uq:expansion` section within the theory portion of the Users Guide
for additional information on the Multilevel PCE algorithm.


*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::

.. code-block::

    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 ensemble
       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.


Theory::

Faq::

See_Also::
