Localized discrete empirical interpolation method

Benjamin Peherstorfer, Daniel Butnaru, Karen Willcox, Hans Joachim Bungartz

Research output: Contribution to journalArticle

Abstract

This paper presents a new approach to construct more efficient reduced-order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method (DEIM). Whereas DEIM projects the nonlinear term onto one global subspace, our localized discrete empirical interpolation method (LDEIM) computes several local subspaces, each tailored to a particular region of characteristic system behavior. Then, depending on the current state of the system, LDEIM selects an appropriate local subspace for the approximation of the nonlinear term. In this way, the dimensions of the local DEIM subspaces, and thus the computational costs, remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes. LDEIM uses machine learning methods in the offline computational phase to discover these regions via clustering. Local DEIM approximations are then computed for each cluster. In the online computational phase, machine-learning-based classification procedures select one of these local subspaces adaptively as the computation proceeds. The classification can be achieved using either the system parameters or a low-dimensional representation of the current state of the system obtained via feature extraction. The LDEIM approach is demonstrated for a reacting flow example of an H2-Air flame. In this example, where the system state has a strong nonlinear dependence on the parameters, the LDEIM provides speedups of two orders of magnitude over standard DEIM.

Original languageEnglish (US)
Pages (from-to)A168-A192
JournalSIAM Journal on Scientific Computing
Volume36
Issue number1
DOIs
StatePublished - 2014

Keywords

  • Clustering
  • Discrete empirical interpolation method
  • Model reduction
  • Nonlinear partial differential equations
  • Proper orthogonal decomposition

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Localized discrete empirical interpolation method'. Together they form a unique fingerprint.

  • Cite this

    Peherstorfer, B., Butnaru, D., Willcox, K., & Bungartz, H. J. (2014). Localized discrete empirical interpolation method. SIAM Journal on Scientific Computing, 36(1), A168-A192. https://doi.org/10.1137/130924408