MANIFOLD APPROXIMATIONS VIA TRANSPORTED SUBSPACES: MODEL REDUCTION FOR TRANSPORT-DOMINATED PROBLEMS

Donsub Rim, Benjamin Peherstorfer, Kyle T. Mandli

Research output: Contribution to journalArticlepeer-review

Abstract

This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.

Original languageEnglish (US)
Pages (from-to)A170-A199
JournalSIAM Journal on Scientific Computing
Volume45
Issue number1
DOIs
StatePublished - 2023

Keywords

  • model reduction
  • nonlinear approximations
  • transport-dominated problems
  • transported subspaces

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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