Compressive sampling methods for sparse polynomial chaos expansions

Abstract

A salient task in uncertainty quantification (UQ) is to study the dependence of a quantity of interest (QoI) on input variables representing system uncertainties. Relying on linear expansions of the QoI in orthogonal polynomial bases of inputs, polynomial chaos expansions (PCEs) are now among the widely used methods in UQ. When there exists a smoothness in the solution being approximated, the PCE exhibits sparsity in that a small fraction of expansion coefficients are significant. By exploiting this sparsity, compressive sampling, also known as compressed sensing, provides a natural framework for accurate PCE using relatively few evaluations of the QoI and in a manner that does not require intrusion into legacy solvers. The PCE possesses a rich structure between the QoI being approximated, the polynomials, and input variables used to perform the approximation and where the QoI is evaluated. In this chapter insights are provided into this structure, summarizing a portion of the current literature on PCE via compressive sampling within the context of UQ.