Localized model reduction for parameterized problems

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

doi:10.1515/9783110671490-006

Localized model reduction for parameterized problems

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

Mathematics of Computational Science

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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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 language	English
Title of host publication	Handbook on Model Order Reduction
Editors	Peter Benner, Stefano Grivet-Talocia, Alfio Quarteroni, Gianluigi Rozza, Wilhelmus Schilders, Luis Miguel Silveira
Publisher	De Gruyter
DOIs	https://doi.org/10.1515/9783110671490-006
Publication status	Published - 2020

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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Buhr2019localizedSubmitted version, 1.87 MB
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Cite this

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title = "Localized model reduction for parameterized problems",

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. ",

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author = "Andreas Buhr and Laura Iapichino and Mario Ohlberger and Stephan Rave and Felix Schindler and Kathrin Smetana",

year = "2020",

doi = "10.1515/9783110671490-006",

language = "English",

editor = "Peter Benner and Stefano Grivet-Talocia and Alfio Quarteroni and Gianluigi Rozza and Wilhelmus Schilders and Silveira, {Luis Miguel}",

booktitle = "Handbook on Model Order Reduction",

publisher = "De Gruyter",

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Localized model reduction for parameterized problems. / Buhr, Andreas; Iapichino, Laura; Ohlberger, Mario et al.

Handbook on Model Order Reduction. ed. / Peter Benner; Stefano Grivet-Talocia; Alfio Quarteroni; Gianluigi Rozza; Wilhelmus Schilders; Luis Miguel Silveira. De Gruyter, 2020.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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T1 - Localized model reduction for parameterized problems

AU - Buhr, Andreas

AU - Iapichino, Laura

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AB - 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.

KW - Localized model reduction

KW - Reduced basis method

KW - Randomized training

KW - A posteriori error estimation

KW - Basis enrichment

KW - Online adaptivity

KW - Parameterized systems

KW - Multi-scale problems

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M3 - Chapter

BT - Handbook on Model Order Reduction

A2 - Benner, Peter

A2 - Grivet-Talocia, Stefano

A2 - Quarteroni, Alfio

A2 - Rozza, Gianluigi

A2 - Schilders, Wilhelmus

A2 - Silveira, Luis Miguel

PB - De Gruyter

ER -

Localized model reduction for parameterized problems

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