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The recovery of approximately sparse or compressible coefficients in aPolynomial Chaos Expansion is a common goal in modern parametric uncertaintyquantification (UQ). However, relatively little effort in UQ has been directedtoward theoretical and computational strategies for addressing the sparsecorruptions problem, where a small number of measurements are highly corrupted.Such a situation has become pertinent today since modern computationalframeworks are sufficiently complex with many interdependent components thatmay introduce hardware and software failures, some of which can be difficult todetect and result in a highly polluted simulation result.  In this paper we present a novel compressive sampling-based theoreticalanalysis for a regularized $\ell^1$ minimization algorithm that aims to recoversparse expansion coefficients in the presence of measurement corruptions. Ourrecovery results are uniform, and prescribe algorithmic regularizationparameters in terms of a user-defined a priori estimate on the ratio ofmeasurements that are believed to be corrupted. We also propose an iterativelyreweighted optimization algorithm that automatically refines the value of theregularization parameter, and empirically produces superior results. Ournumerical results test our framework on several medium-to-high dimensionalexamples of solutions to parameterized differential equations, and demonstratethe effectiveness of our approach.

Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations