research-article

Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates

Authors:

Benjamin Peherstorfer,

Karen WillcoxAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 37, Issue 4

Pages A2123 - A2150

https://doi.org/10.1137/140989169

Published: 01 January 2015 Publication History

Abstract

This work presents a nonlinear model reduction approach for systems of equations stemming from the discretization of partial differential equations with nonlinear terms. Our approach constructs a reduced system with proper orthogonal decomposition and the discrete empirical interpolation method (DEIM); however, whereas classical DEIM derives a linear approximation of the nonlinear terms in a static DEIM space generated in an offline phase, our method adapts the DEIM space as the online calculation proceeds and thus provides a nonlinear approximation. The online adaptation uses new data to produce a reduced system that accurately approximates behavior not anticipated in the offline phase. These online data are obtained by querying the full-order system during the online phase, but only at a few selected components to guarantee a computationally efficient adaptation. Compared to the classical static approach, our online adaptive and nonlinear model reduction approach achieves accuracy improvements of up to three orders of magnitude in our numerical experiments with time-dependent and steady-state nonlinear problems. The examples also demonstrate that through adaptivity, our reduced systems provide valid approximations of the full-order systems outside of the parameter domains for which they were initially built in the offline phase.

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 37, Issue 4

DOI:10.1137/sjoce3.37.4

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© 2015, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2015

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Cited By

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Bruna JPeherstorfer BVanden-Eijnden E(2024)Neural Galerkin schemes with active learning for high-dimensional evolution equationsJournal of Computational Physics10.1016/j.jcp.2023.112588496:COnline publication date: 27-Feb-2024
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