On Polynomial Chaos Expansion via Gradient-enhanced $\ell_1$-minimization

TL;DR

This paper explores gradient-enhanced $ ext{l}_1$-minimization for Polynomial Chaos Expansions, demonstrating improved stability and convergence through probabilistic analysis and numerical examples, leading to more efficient uncertainty quantification.

Contribution

It provides the first stability and convergence analysis for gradient-enhanced $ ext{l}_1$-minimization in PCE, showing derivative info improves recovery conditions.

Findings

Derivative information improves measurement matrix properties.

Including derivatives reduces computational cost for PCE recovery.

Numerical examples confirm theoretical improvements in solution recovery.

Abstract

Gradient-enhanced Uncertainty Quantification (UQ) has received recent attention, in which the derivatives of a Quantity of Interest (QoI) with respect to the uncertain parameters are utilized to improve the surrogate approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and when the QoI can be represented by a sparse PCE, $\ell_1$-minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient-enhanced $\ell_1$-minimization, where derivative information is computed to accelerate the identification of the PCE coefficients. For this approach, stability and convergence analysis are lacking, and thus we address these here with a probabilistic result. In particular, with an appropriate normalization, we show the inclusion of derivative information will almost-surely lead to improved conditions, e.g. related to the null-space and coherence of the measurement matrix, for a successful solution recovery. Further, we demonstrate our analysis empirically via three numerical examples: a manufactured PCE, an elliptic partial differential equation with random inputs, and a plane Poiseuille flow with random boundaries. These examples all suggest that including derivative information admits solution recovery at reduced computational cost.