A mixed ℓ₁ regularization approach for sparse simultaneous approximation of parameterized PDEs

¹ Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
² Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6164, USA
³ Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

^* Corresponding author: cwebst13@utk.edu; webstercg@ornl.gov

Received: 15 December 2018
Accepted: 27 June 2019

Abstract

We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based ℓ₁ regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best s-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.