Blurb:: Sample allocation based on restricted isometry property (RIP) within multilevel polynomial chaos

Description:: 
Multilevel polynomial chaos with compressed sensing may allocate the
number of samples per level based on the restricted isometry property
(RIP), applied to recovery at level \f$l\f$:

\f[ N_l ~ \geq ~ s_l ~ log^3(s_l) ~ L_l ~ log(C_l) \f]

for sparsity \f$s\f$, cardinality \f$C\f$, and mutual coherence \f$L\f$.
The adaptive algorithm starts from a pilot sample, shapes the profile
based on observed sparsity, and iterates until convergence.  In practice,
RIP sampling levels are quite conservative, and a collocation ratio
constraint (\f$N_l \leq r C_l\f$, where \f$r\f$ defaults to 2) must be
enforced on the profile.

The algorithm relies on observed sparsity, it is appropriate for use
with regularized solvers for compressed sensing.  It employs
orthogonal matching pursuit (OMP) by default and automatically
activates cross-validation in order to choose the best noise parameter
value for the recovery.

This capability is \b experimental.  Sample allocation by greedy
refinement is generally preferred.

Topics::

Examples::

This example starts with sparse recovery for a second-order candidate
expansion at each level.  As the number of samples is adapted for each
level, as dictated by the number of sparse coefficient sets recovered
for each level, the candidate expansion order is incremented as needed
in order to synchronize with the specified collocation ratio.

\verbatim
method,
	model_pointer = 'HIERARCH'
	multilevel_polynomial_chaos
	  orthogonal_matching_pursuit
	  expansion_order_sequence = 2
	  pilot_samples = 10
	  collocation_ratio = .9
	  allocation_control rip_sampling
	  seed = 1237
	  convergence_tolerance = .01
\endverbatim

Theory::

Faq::
See_Also::
