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Localized model reduction for parameterized problems
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In detail, it is shown how optimal local approximation spaces can be constructed and approximated by random sampling and an overview of possible conforming and non-conforming couplings of the local spaces is provided.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.
This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order Reduction. Walter De Gruyter GmbH, Berlin, 2019+.read more
Citations
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Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations
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Towards Automatic and Reliable Localized Model Order Reduction.
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References
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Journal ArticleDOI
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
Journal ArticleDOI
Coupling of substructures for dynamic analyses.
M. C. C. Bampton,Roy R. Craig +1 more
TL;DR: In this article, a method for treating a complex structure as an assemblage of distinct regions, or substructures, is presented using basic mass and stiffness matrices, together with conditions of geometrical compatibility along substructure boundaries.
Journal ArticleDOI
The Partition of Unity Method
Ivo Babuška,Jens Markus Melenk +1 more
TL;DR: In this article, a new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved, which can therefore be more efficient than the usual finite element methods.
Posted Content
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
TL;DR: In this article, a modular framework for constructing randomized algorithms that compute partial matrix decompositions is presented, which uses random sampling to identify a subspace that captures most of the action of a matrix and then the input matrix is compressed to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization.