Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations
For uncertainty quantification practitioners, this provides a fault-tolerant approach to handle corrupted simulation data, though the improvements are incremental over existing compressed sensing methods.
The paper addresses the recovery of sparse Polynomial Chaos Expansion coefficients in the presence of sparse measurement corruptions, presenting a compressive sampling-based analysis for a regularized ℓ1 minimization algorithm with uniform recovery guarantees and an iteratively reweighted optimization method that improves empirical performance.
The recovery of approximately sparse or compressible coefficients in a Polynomial Chaos Expansion is a common goal in modern parametric uncertainty quantification (UQ). However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized $\ell^1$ minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform, and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.