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

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. 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.

Authors:
 [1];  [1];  [2]; ORCiD logo [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)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1479490
Report Number(s):
SAND-2018-10441J
Journal ID: ISSN 2166-2525; 668226
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM/ASA Journal on Uncertainty Quantification
Additional Journal Information:
Journal Volume: 6; Journal Issue: 4; Journal ID: ISSN 2166-2525
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; compressed sensing; corrupted measurements; fault tolerance

Citation Formats

Adcock, Ben, Bao, Anyi, Jakeman, John Davis, and Narayan, Akil. Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations. United States: N. p., 2018. Web. doi:10.1137/17M112590X.
Adcock, Ben, Bao, Anyi, Jakeman, John Davis, & Narayan, Akil. Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations. United States. doi:10.1137/17M112590X.
Adcock, Ben, Bao, Anyi, Jakeman, John Davis, and Narayan, Akil. Tue . "Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations". United States. doi:10.1137/17M112590X.
@article{osti_1479490,
title = {Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations},
author = {Adcock, Ben and Bao, Anyi and Jakeman, John Davis and Narayan, Akil},
abstractNote = {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.},
doi = {10.1137/17M112590X},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
issn = {2166-2525},
number = 4,
volume = 6,
place = {United States},
year = {2018},
month = {10}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on October 16, 2019
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