Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points

TL;DR

This paper analyzes the stability of empirical interpolation methods in nonlinear model reduction, demonstrating that randomized sampling with oversampling enhances stability, and proposes a deterministic sampling strategy for improved accuracy.

Contribution

It provides a probabilistic analysis of empirical interpolation stability and introduces a deterministic sampling method to reduce approximation errors.

Findings

Randomized sampling with oversampling ensures stable approximations.

Gappy POD with oversampling is more stable than standard empirical interpolation.

Numerical experiments confirm the stability advantages of the proposed methods.

Abstract

This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state field approximation from measurements. Empirical interpolation derives approximations from a few samples (measurements) via interpolation in low-dimensional spaces. It has been observed that empirical interpolation can become unstable if the samples are perturbed due to, e.g., noise, turbulence, and numerical inaccuracies. The main contribution of this work is a probabilistic analysis that shows that stable approximations are obtained if samples are randomized and if more samples than dimensions of the low-dimensional spaces are used. Oversampling, i.e., taking more sampling points than dimensions of the low-dimensional spaces, leads to approximations via regression and is known under the name of gappy proper orthogonal decomposition. Building on the insights of the probabilistic analysis, a deterministic sampling strategy is presented that aims to achieve lower approximation errors with fewer points than randomized sampling by taking information about the low-dimensional spaces into account. Numerical results of reconstructing velocity fields from noisy measurements of combustion processes and model reduction in the presence of noise demonstrate the instability of empirical interpolation and the stability of gappy proper orthogonal decomposition with oversampling.