Localized model reduction for parameterized problems

Andreas Buhr, Laura Iapichino, Mario Ohlberger, Stephan Rave, Felix Schindler, Kathrin Smetana

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    Abstract

    In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that have only support on part of the domain and compute a global approximation by a suitable coupling of the local spaces. In detail, we show how optimal local approximation spaces can be constructed and approximated by random sampling. An overview of possible conforming and non-conforming couplings of the local spaces is provided and corresponding localized a posteriori error estimates are derived. We introduce concepts of local basis enrichment, which includes a discussion of adaptivity. Implementational aspects of localized model reduction methods are addressed.
    Finally, we illustrate the presented concepts for multiscale, linear elasticity and fluid-flow problems, providing several numerical experiments.
    Original languageEnglish
    Title of host publicationHandbook on Model Order Reduction
    EditorsPeter Benner, Stefano Grivet-Talocia, Alfio Quarteroni, Gianluigi Rozza, Wilhelmus Schilders, Luis Miguel Silveira
    PublisherWalter De Gruyter
    Publication statusAccepted/In press - 2019

    Keywords

    • Localized model reduction
    • Reduced basis method
    • Randomized training
    • A posteriori error estimation
    • Basis enrichment
    • Online adaptivity
    • Parameterized systems
    • Multi-scale problems

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  • Cite this

    Buhr, A., Iapichino, L., Ohlberger, M., Rave, S., Schindler, F., & Smetana, K. (Accepted/In press). Localized model reduction for parameterized problems. In P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. Schilders, & L. M. Silveira (Eds.), Handbook on Model Order Reduction Walter De Gruyter.