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

TL;DR

This paper investigates compressive sensing techniques for constructing sparse polynomial chaos expansions in high-dimensional uncertainty quantification, introducing cross-validation and stop-sampling strategies to improve accuracy and efficiency.

Contribution

It develops automated regularization and stopping strategies for compressive sensing solvers, providing practical guidelines and demonstrating effectiveness on complex, high-dimensional physical models.

Findings

ADMM outperforms other solvers in accuracy and speed

Cross-validation effectively mitigates overfitting in sparse recovery

Method successfully applied to a 24-dimensional turbulent jet-in-crossflow simulation

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 of 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 tradeoffs 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.