rip_sampling

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

Specification

• Alias: None

• Arguments: None

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 $l$ :

$$N_l\ge s_{l}lo{g}^3(s_l)L_{l}log(C_l)$$

for sparsity $s$ , cardinality $C$ , and mutual coherence $L$ . 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 ( $N_l\le r{C}_l$ , where $r$ 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.

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.

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