  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.
