Stabilizing discrete empirical interpolation via randomized and deterministic oversampling

This work investigates randomized and deterministic oversampling in (discrete) empirical interpolation for nonlinear model reduction. Empirical interpolation derives approximations of nonlinear terms from a few samples via interpolation in low-dimensional spaces. It has been demonstrated that empirical interpolation can become unstable if the samples from the nonlinear terms are perturbed due to, e.g., noise, turbulence, and numerical inaccuracies. We demonstrate with a probabilistic analysis that randomized oversampling stabilizes empirical interpolation in the presence of noise. Furthermore, we discuss deterministic oversampling strategies that select points by descending in directions of eigenvectors corresponding to sampling point updates and by establishing connections between sampling point selection and clustering. Numerical experiments with synthetic and diffusion-reaction problems demonstrate the stability of oversampled empirical interpolation in the presence of noise.

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