TY - JOUR
T1 - Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points
AU - Peherstorfer, Benjamin
AU - Drmac, Zlatko
AU - Gugercin, Serkan
N1 - Publisher Copyright:
© 2020 Benjamin Peherstorfer, Serkan Gugercin, Zlatko Drmac
PY - 2020
Y1 - 2020
N2 - This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state feld 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 felds 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.
AB - This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state feld 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 felds 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.
KW - Empirical interpolation
KW - Gappy proper orthogonal decomposition
KW - Model reduction
KW - Noisy observations
KW - Nonlinear model reduction
KW - Randomized model reduction
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U2 - 10.1137/19M1307391
DO - 10.1137/19M1307391
M3 - Article
AN - SCOPUS:85093107106
SN - 1064-8275
VL - 42
SP - A2837-A2864
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 5
M1 - 19M1307391
ER -