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

### Abstract

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.

Type
Publication
SIAM/ASA Journal on Uncertainty Quantification
Date
Citation
X. Huan, C. Safta, K. Sargsyan, Z. P. Vane, G. Lacaze, J. C. Oefelein, and H. N. Najm. Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions. SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, No. 2, pp. 907–936, 2018. https://dx.doi.org/10.1137/17M1141096

### BibTeX

@article{Huan2018_sjuq,
author = {Huan, Xun and Safta, Cosmin and Sargsyan, Khachik and Vane, Zachary P. and Lacaze, Guilhem and Oefelein, Joseph C. and Najm, Habib N.},
doi = {10.1137/17M1141096},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = {2},
pages = {907--936},
title = {{Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions}},
volume = {6},
year = {2018}
}