SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2837-A2864, January 2020.
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.

中文翻译：

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具有随机和确定性采样点的离散经验插值和Gappy正确正交分解的稳定性

SIAM科学计算杂志，第42卷，第5期，第A2837-A2864页，2020年1月。
这项工作研究了（离散）经验插值的稳定性，以减少非线性模型并从测量中近似出状态场。经验插值是在低维空间中通过插值从一些样本（测量值）得出近似值。已经观察到，如果样本由于例如噪声，湍流和数值误差而受到干扰，则经验插值会变得不稳定。这项工作的主要贡献是概率分析，该分析表明，如果将样本随机化并且使用的样本多于低维空间的维度，则可以获得稳定的近似值。过采样（即，比低维空间的维获取更多的采样点）会导致通过回归进行逼近，并且以“空洞的适当正交分解”的名称而闻名。在概率分析的基础上，提出了确定性采样策略，该策略旨在通过考虑有关低维空间的信息，以比随机采样少的点数实现更低的逼近误差。从燃烧过程的噪声测量和存在噪声的模型简化中重建速度场的数值结果表明，经验插值的不稳定性以及带有过采样的空洞适当正交分解的稳定性。