Blurb:: Sample allocation based on greedy refinement within multifidelity stochastic collocation

Description::

Multifidelity stochastic collocation supports greedy refinement
strategies using tensor and sparse grids for both nodal and
hierarchical collocation approaches.  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 an integrated competition, and the candidate that induces the
largest change in the statistical QoI (response covariance by default,
or results of any 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 stochastic collocation
using nodel interpolation starts from a zeroth-order expansion (a
constant) for each level, and generates uniform candidate refinements
for each level that are competed in a greedy competition.  The number
of new samples for the incremented candidate expansion order is
determined from the quadrature rules of the new sparse grid level.  In
this case, the number of candidates for each level is limited to one
uniform refinement of the current sparse grid level.

\verbatim
method,
	model_pointer = 'HIERARCH'
	multifidelity_stoch_collocation
	  nodal
	  allocation_control greedy
	  p_refinement uniform
	    sparse_grid_level_sequence = 0 unrestricted
	    convergence_tolerance 1.e-3
\endverbatim

The next example employs generalized sparse grids and hierarchical
interpolation.  Each 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 levels is competed within a
unified 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 level and the number of candidates grows
rapidly with random dimension and grid level.

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

Theory::

Faq::
See_Also::
