Abstract

The recovery of approximately sparse or compressible coefficients in a polynomial chaos expansion is a common goal in many modern parametric uncertainty quantification (UQ) problems. 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 (the theoretical guarantees hold for all compressible signals and compressible corruptions vectors) 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.

Keywords

  1. compressed sensing
  2. corrupted measurements
  3. fault tolerance

MSC codes

  1. 42A10
  2. 41A10
  3. 65D15

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Information & Authors

Information

Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 1424 - 1453
ISSN (online): 2166-2525

History

Submitted: 17 April 2017
Accepted: 27 August 2018
Published online: 16 October 2018

Keywords

  1. compressed sensing
  2. corrupted measurements
  3. fault tolerance

MSC codes

  1. 42A10
  2. 41A10
  3. 65D15

Authors

Affiliations

Funding Information

Defense Advanced Research Projects Agency https://doi.org/10.13039/100000185 : N660011524053
Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-15-1-0467
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : 611675
Division of Mathematical Sciences https://doi.org/10.13039/100000121 : DMS-1552238

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