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Article: Sparse generalized multiscale finite element methods and their applications

TitleSparse generalized multiscale finite element methods and their applications
Authors
KeywordsMultiscale finite element
Sparsity
Multiscale model reduction
L minimization 1
Issue Date2016
Citation
International Journal for Multiscale Computational Engineering, 2016, v. 14, n. 1, p. 1-23 How to Cite?
Abstract© 2016 by Begell House, Inc. In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.
Persistent Identifierhttp://hdl.handle.net/10722/287051
ISSN
2020 Impact Factor: 1.508
2020 SCImago Journal Rankings: 0.430
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorChung, Eric-
dc.contributor.authorEfendiev, Yalchin-
dc.contributor.authorLeung, Wing Tat-
dc.contributor.authorLi, Guanglian-
dc.date.accessioned2020-09-07T11:46:21Z-
dc.date.available2020-09-07T11:46:21Z-
dc.date.issued2016-
dc.identifier.citationInternational Journal for Multiscale Computational Engineering, 2016, v. 14, n. 1, p. 1-23-
dc.identifier.issn1543-1649-
dc.identifier.urihttp://hdl.handle.net/10722/287051-
dc.description.abstract© 2016 by Begell House, Inc. In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.-
dc.languageeng-
dc.relation.ispartofInternational Journal for Multiscale Computational Engineering-
dc.subjectMultiscale finite element-
dc.subjectSparsity-
dc.subjectMultiscale model reduction-
dc.subjectL minimization 1-
dc.titleSparse generalized multiscale finite element methods and their applications-
dc.typeArticle-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1615/IntJMultCompEng.2015014280-
dc.identifier.scopuseid_2-s2.0-84963776426-
dc.identifier.volume14-
dc.identifier.issue1-
dc.identifier.spage1-
dc.identifier.epage23-
dc.identifier.isiWOS:000374274000001-
dc.identifier.issnl1543-1649-

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