  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.
