A preconditioning approach for improved estimation of sparse polynomial chaos expansions

TL;DR

This paper introduces a preconditioning method for sparse polynomial chaos expansions that enhances measurement matrix properties, leading to improved recovery accuracy without compromising signal-to-noise ratio.

Contribution

It proposes a novel preconditioning scheme that balances incoherence improvement and noise preservation for better polynomial chaos estimation.

Findings

Preconditioning improves measurement matrix incoherence.

The scheme enhances accuracy of polynomial chaos expansions.

Numerical results confirm theoretical benefits.

Abstract

Compressive sampling has been widely used for sparse polynomial chaos (PC) approximation of stochastic functions. The recovery accuracy of compressive sampling highly depends on the incoherence properties of the measurement matrix. In this paper, we consider preconditioning the underdetermined system of equations that is to be solved. Premultiplying a linear equation system by a non-singular matrix results in an equivalent equation system, but it can potentially improve the incoherence properties of the resulting preconditioned measurement matrix and lead to a better recovery accuracy. When measurements are noisy, however, preconditioning can also potentially result in a worse signal-to-noise ratio, thereby deteriorating recovery accuracy. In this work, we propose a preconditioning scheme that improves the incoherence properties of measurement matrix and at the same time prevents undesirable deterioration of signal-to-noise ratio. We provide theoretical motivations and numerical examples that demonstrate the promise of the proposed approach in improving the accuracy of estimated polynomial chaos expansions.