TY - CHAP
T1 - Localized model reduction for parameterized problems
AU - Buhr, Andreas
AU - Iapichino, Laura
AU - Ohlberger, Mario
AU - Rave, Stephan
AU - Schindler, Felix
AU - Smetana, Kathrin
N1 - Publisher Copyright:
© 2021 Gianluigi Rozza et al.
PY - 2020/12/16
Y1 - 2020/12/16
N2 - 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 nonconforming 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.
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 nonconforming 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 - A posteriori error estimation
KW - Basis enrichment
KW - Localized model reduction
KW - Multiscale problems
KW - Online adaptivity
KW - Parameterized systems
KW - Randomized training
KW - Reduced basis method
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U2 - 10.1515/9783110671490-006
DO - 10.1515/9783110671490-006
M3 - Chapter
AN - SCOPUS:85109227361
SN - 9783110671407
SP - 245
EP - 305
BT - Snapshot-Based Methods and Algorithms
ER -