Online adaptive model reduction for nonlinear systems via low-rank updates

Benjamin Peherstorfer, Karen Willcox

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)A2123-A2150
JournalSIAM Journal on Scientific Computing
Volume37
Issue number4
DOIs
StatePublished - Jan 1 2015

Fingerprint

Model Reduction
Interpolation Method
Nonlinear systems
Interpolation
Nonlinear Systems
Update
Nonlinear Model
Nonlinear Approximation
Orthogonal Decomposition
Partial differential equations
Adaptivity
Linear Approximation
Term
System of equations
Nonlinear Problem
Decomposition
Partial differential equation
Discretization
Numerical Experiment
Valid

Keywords

  • Adaptive model reduction
  • Empirical interpolation
  • Nonlinear systems
  • Proper orthogonal decomposition

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

Online adaptive model reduction for nonlinear systems via low-rank updates. / Peherstorfer, Benjamin; Willcox, Karen.

In: SIAM Journal on Scientific Computing, Vol. 37, No. 4, 01.01.2015, p. A2123-A2150.

Research output: Contribution to journalArticle

Peherstorfer, Benjamin ; Willcox, Karen. / Online adaptive model reduction for nonlinear systems via low-rank updates. In: SIAM Journal on Scientific Computing. 2015 ; Vol. 37, No. 4. pp. A2123-A2150.
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