Abstract

Co-authors: Hoang Tran (Oak Ridge National Laboratory) & Clayton Webster (University of Tennessee & Oak Ridge National Laboratory) We present and analyze a novel sparse polynomial approximation method for the solution of PDEs with stochastic and parametric inputs. Our approach treats the parameterized problem as a problem of joint-sparse signal recovery, i.e., simultaneous reconstruction of a set of sparse signals, sharing a common sparsity pattern, from a countable, possibly infinite, set of measurements. In this setting, the support set of the signal is assumed to be unknown and the measurements may be corrupted by noise. We propose the solution of a linear inverse problem via convex sparse regularization for an approximation to the true signal. Our approach allows for global approximations of the solution over both physical and parametric domains. In addition, we show that the method enjoys the minimal sample complexity requirements common to compressed sensing-based approaches. We then perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the recovery properties of the proposed approach.