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Article: Proper Orthogonal Decomposition Method for Multiscale Elliptic PDEs with Random Coefficients

TitleProper Orthogonal Decomposition Method for Multiscale Elliptic PDEs with Random Coefficients
Authors
KeywordsRandom partial differential equations (RPDEs)
Uncertainty quantification (UQ)
Proper orthogonal decomposition (POD) method
Multiscale reduced basis
High-contrast problem
Issue Date2020
PublisherElsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/cam
Citation
Journal of Computational and Applied Mathematics, 2020, v. 370, p. article no. 112635 How to Cite?
AbstractIn this paper, we develop an efficient multiscale reduced basis method to solve elliptic PDEs with multiscale and random coefficients in a multi-query setting. Our method consists of offline and online stages. In the offline stage, a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition (POD) method. Moreover, local tensor spaces are defined to approximate the solution space of the multiscale random PDEs. In the online stage, a weak formulation is derived and discretized using the Galerkin method to compute the solution. Since the multiscale reduced basis functions can efficiently approximate the high-dimensional solution space, our method is very efficient in solving multiscale elliptic PDEs with random coefficients. Convergence analysis of the proposed method is presented, which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the POD method. Finally, numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space.
Persistent Identifierhttp://hdl.handle.net/10722/280955
ISSN
2019 Impact Factor: 2.037
2015 SCImago Journal Rankings: 1.089

 

DC FieldValueLanguage
dc.contributor.authorMA, D-
dc.contributor.authorChing, W-K-
dc.contributor.authorZhang, Z-
dc.date.accessioned2020-02-25T07:43:14Z-
dc.date.available2020-02-25T07:43:14Z-
dc.date.issued2020-
dc.identifier.citationJournal of Computational and Applied Mathematics, 2020, v. 370, p. article no. 112635-
dc.identifier.issn0377-0427-
dc.identifier.urihttp://hdl.handle.net/10722/280955-
dc.description.abstractIn this paper, we develop an efficient multiscale reduced basis method to solve elliptic PDEs with multiscale and random coefficients in a multi-query setting. Our method consists of offline and online stages. In the offline stage, a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition (POD) method. Moreover, local tensor spaces are defined to approximate the solution space of the multiscale random PDEs. In the online stage, a weak formulation is derived and discretized using the Galerkin method to compute the solution. Since the multiscale reduced basis functions can efficiently approximate the high-dimensional solution space, our method is very efficient in solving multiscale elliptic PDEs with random coefficients. Convergence analysis of the proposed method is presented, which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the POD method. Finally, numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space.-
dc.languageeng-
dc.publisherElsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/cam-
dc.relation.ispartofJournal of Computational and Applied Mathematics-
dc.subjectRandom partial differential equations (RPDEs)-
dc.subjectUncertainty quantification (UQ)-
dc.subjectProper orthogonal decomposition (POD) method-
dc.subjectMultiscale reduced basis-
dc.subjectHigh-contrast problem-
dc.titleProper Orthogonal Decomposition Method for Multiscale Elliptic PDEs with Random Coefficients-
dc.typeArticle-
dc.identifier.emailChing, W-K: wching@hku.hk-
dc.identifier.emailZhang, Z: zhangzw@hku.hk-
dc.identifier.authorityChing, W-K=rp00679-
dc.identifier.authorityZhang, Z=rp02087-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1016/j.cam.2019.112635-
dc.identifier.scopuseid_2-s2.0-85076099548-
dc.identifier.hkuros309248-
dc.identifier.volume370-
dc.identifier.spagearticle no. 112635-
dc.identifier.epagearticle no. 112635-
dc.publisher.placeNetherlands-

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