  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.
