A gradient enhanced $\ell_1$-minimization for sparse approximation of polynomial chaos expansions

TL;DR

This paper introduces a gradient-enhanced $ ext{l}_1$-minimization method for efficiently constructing sparse polynomial chaos expansions by incorporating gradient information, leading to more stable and accurate coefficient recovery.

Contribution

The paper presents a novel gradient-enhanced $ ext{l}_1$-minimization framework with preconditioning techniques that improve sparse polynomial chaos expansion recovery, applicable to bounded and unbounded domains.

Findings

Gradient-enhanced approach improves sparse recovery accuracy.

Preconditioning ensures stability and robustness.

Method reduces computational cost compared to standard $ ext{l}_1$-minimization.

Abstract

We investigate a gradient-enhanced $\ell_1$-minimization for constructing sparse polynomial chaos expansions. In addition to function evaluations, measurements of the function gradient is also included to accelerate the identification of expansion coefficients. By designing appropriate preconditioners to the measurement matrix, we show gradient-enhanced $\ell_1$ minimization leads to stable and accurate coefficient recovery. The framework for designing preconditioners is quite general and it applies to recover of functions whose domain is bounded or unbounded. Comparisons between the gradient enhanced approach and the standard $\ell_1$-minimization are also presented and numerical examples suggest that the inclusion of derivative information can guarantee sparse recovery at a reduced computational cost.