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Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations

Journal Article · · SIAM/ASA Journal on Uncertainty Quantification
DOI:https://doi.org/10.1137/17M112590X· OSTI ID:1479490
 [1];  [1];  [2];  [3]
  1. Simon Fraser Univ., Burnaby, BC (Canada)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  3. Univ. of Utah, Salt Lake City, UT (United States)
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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. Lastly, 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.

Research Organization:
Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC04-94AL85000
OSTI ID:
1479490
Report Number(s):
SAND--2018-10441J; 668226
Journal Information:
SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 4 Vol. 6; ISSN 2166-2525
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

Figures / Tables (8)

Table 1(p. 6)table Table 1
Figure 1(p. 18)figure Figure 1
Figure 2(p. 19)figure Figure 2
Figure 3(p. 20)figure Figure 3
Figure 4(p. 21)figure Figure 4
Figure 5(p. 23)figure Figure 5
Figure 6(p. 24)figure Figure 6
Figure 7(p. 24)figure Figure 7

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Related Subjects

97 MATHEMATICS AND COMPUTING
compressed sensing
corrupted measurements
fault tolerance