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Title: Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions

Abstract

Here, compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational trade-offs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.

Authors:
ORCiD logo [1];  [1];  [1];  [1];  [1];  [1];  [1]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
Defense Advanced Research Projects Agency (DARPA); USDOE
OSTI Identifier:
1459932
Alternate Identifier(s):
OSTI ID: 1459941; OSTI ID: 1478334
Report Number(s):
SAND2018-6855J; SAND-2018-6855J; SAND-2017-8048J
Journal ID: ISSN 2166-2525; 664904
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
SIAM/ASA Journal on Uncertainty Quantification
Additional Journal Information:
Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 2166-2525
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; uncertainty quantification; LASSO; compressed sensing; sparse regression; sparse reconstruction; sequential compressive sensing

Citation Formats

Huan, Xun, Safta, Cosmin, Sargsyan, Khachik, Vane, Zachary P., Lacaze, Guilhem, Oefelein, Joseph C., and Najm, Habib N. Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions. United States: N. p., 2018. Web. doi:10.1137/17M1141096.
Huan, Xun, Safta, Cosmin, Sargsyan, Khachik, Vane, Zachary P., Lacaze, Guilhem, Oefelein, Joseph C., & Najm, Habib N. Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions. United States. doi:10.1137/17M1141096.
Huan, Xun, Safta, Cosmin, Sargsyan, Khachik, Vane, Zachary P., Lacaze, Guilhem, Oefelein, Joseph C., and Najm, Habib N. Tue . "Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions". United States. doi:10.1137/17M1141096. https://www.osti.gov/servlets/purl/1459932.
@article{osti_1459932,
title = {Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions},
author = {Huan, Xun and Safta, Cosmin and Sargsyan, Khachik and Vane, Zachary P. and Lacaze, Guilhem and Oefelein, Joseph C. and Najm, Habib N.},
abstractNote = {Here, compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational trade-offs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.},
doi = {10.1137/17M1141096},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = 2,
volume = 6,
place = {United States},
year = {2018},
month = {6}
}

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