Compressive Sensing with CrossValidation and StopSampling 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 highdimensional 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 crossvalidation, and a heuristic strategy to guide the stopsampling 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 jetincrossflow involving a 24dimensional input. Through empirical phasetransition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational tradeoffs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, highdimensional 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:

 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLCA), Livermore, CA (United States); Sandia National Lab. (SNLNM), 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):
 SAND20186855J; SAND20186855J; SAND20178048J
Journal ID: ISSN 21662525; 664904
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 SIAM/ASA Journal on Uncertainty Quantification
 Additional Journal Information:
 Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 21662525
 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 CrossValidation and StopSampling 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 CrossValidation and StopSampling 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 CrossValidation and StopSampling 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 CrossValidation and StopSampling 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 highdimensional 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 crossvalidation, and a heuristic strategy to guide the stopsampling 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 jetincrossflow involving a 24dimensional input. Through empirical phasetransition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational tradeoffs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, highdimensional 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},
issn = {21662525},
number = 2,
volume = 6,
place = {United States},
year = {2018},
month = {6}
}
Web of Science
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