Blurb:: Sample allocation based on greedy refinement within multifidelity polynomial chaos

Description::
Multifidelity polynomial chaos supports greedy
refinement strategies, spanning regression and projection approaches
for computing expansion coefficients.  The key idea is that each level
of the model hierarchy being approximated can generate one or more
candidates for refinement.  These candidates are competed against each
other within a unified competition, and the candidate that induces the
largest change in the statistical QoI (response covariance by default,
or results of any \f$z/p/\beta/\beta^*\f$ level mappings when specified),
normalized by relative cost of evaluating the candidate, is selected
and then used to generate additional candidates for consideration at
its model level.

Topics::

Examples::

The following example of greedy multifidelity regression starts from a
zeroth-order reference expansion (a constant) for each level, and
generates candidate refinements for each level that are competed in an
integrated greedy competition.  The number of new samples for the
incremented candidate expansion order is determined from the collocation
ratio.  In this case, the number of candidates for each level is limited
to one uniform refinement of the current expansion order.

\verbatim
method,
	model_pointer = 'HIERARCH'
	multifidelity_polynomial_chaos
	  allocation_control greedy
	  p_refinement uniform
	    expansion_order_sequence = 0
	    collocation_ratio = .9  seed = 160415
	    orthogonal_matching_pursuit
	    convergence_tolerance 1.e-2
\endverbatim

The next example employs generalized sparse grids within a greedy
multifidelity competition.  Each modeling level starts from a level 0
reference grid (a single point) and generates multiple admissible
index set candidates.  The full set of candidates across all model
levels is competed within an integrated greedy competition, where the
greedy selection metric is the induced change in the statistical QoI,
normalized by the aggregate simulation cost of the index set
candidate.  In this case, there are multiple candidates for each model
level and the number of candidates grows rapidly with random dimension
and grid level.

\verbatim
method,
	model_pointer = 'HIERARCH'
	multifidelity_polynomial_chaos
	  allocation_control greedy
	  p_refinement dimension_adaptive generalized
	    sparse_grid_level_sequence = 0 unrestricted
	    convergence_tolerance 1.e-8
\endverbatim


Theory::

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