A general framework of rotational sparse approximation in uncertainty quantification

TL;DR

This paper introduces a flexible framework for improving uncertainty quantification by applying rotational sparse approximation to generalized polynomial chaos, enhancing the sparsity and accuracy of the representation.

Contribution

It develops a novel rotational approach combined with nonconvex regularizations and ADMM optimization to achieve sparser gPC expansions in uncertainty quantification.

Findings

Nonconvex regularizations outperform l1 in sparse recovery.

Rotational approach improves the sparsity of gPC coefficients.

Numerical results demonstrate superior accuracy with the proposed method.

Abstract

This paper proposes a general framework to estimate coefficients of generalized polynomial chaos (gPC) used in uncertainty quantification via rotational sparse approximation. In particular, we aim to identify a rotation matrix such that the gPC expansion of a set of random variables after the rotation has a sparser representation. However, this rotational approach alters the underlying linear system to be solved, which makes finding the sparse coefficients more difficult than the case without rotation. To solve this problem, we examine several popular nonconvex regularizations in compressive sensing (CS) that perform better than the classic l1 approach empirically. All these regularizations can be minimized by the alternating direction method of multipliers (ADMM). Numerical examples show superior performance of the proposed combination of rotation and nonconvex sparse promoting regularizations over the ones without rotation and with rotation but using the convex l1 approach.