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Article: A multiscale data-driven stochastic method for elliptic PDEs with random coefficients

TitleA multiscale data-driven stochastic method for elliptic PDEs with random coefficients
Authors
KeywordsModel reduction
Karhunen-Loè
Data-driven methods
Ve expansion
Uncertainty quantification
Stochastic partial differential equations
Multiscale problems
Issue Date2015
Citation
Multiscale Modeling and Simulation, 2015, v. 13, n. 1, p. 173-204 How to Cite?
Abstract© 2015 Society for Industrial and Applied Mathematics. In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449-474] and the data-driven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452-493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen-Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.
Persistent Identifierhttp://hdl.handle.net/10722/219845
ISSN
2015 Impact Factor: 1.585
2015 SCImago Journal Rankings: 1.062

 

DC FieldValueLanguage
dc.contributor.authorZhang, Zhiwen-
dc.contributor.authorCi, Maolin-
dc.contributor.authorHou, Thomas Y.-
dc.date.accessioned2015-09-23T02:58:05Z-
dc.date.available2015-09-23T02:58:05Z-
dc.date.issued2015-
dc.identifier.citationMultiscale Modeling and Simulation, 2015, v. 13, n. 1, p. 173-204-
dc.identifier.issn1540-3459-
dc.identifier.urihttp://hdl.handle.net/10722/219845-
dc.description.abstract© 2015 Society for Industrial and Applied Mathematics. In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449-474] and the data-driven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452-493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen-Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.-
dc.languageeng-
dc.relation.ispartofMultiscale Modeling and Simulation-
dc.subjectModel reduction-
dc.subjectKarhunen-Loè-
dc.subjectData-driven methods-
dc.subjectVe expansion-
dc.subjectUncertainty quantification-
dc.subjectStochastic partial differential equations-
dc.subjectMultiscale problems-
dc.titleA multiscale data-driven stochastic method for elliptic PDEs with random coefficients-
dc.typeArticle-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.doi10.1137/130948136-
dc.identifier.scopuseid_2-s2.0-84925068983-
dc.identifier.volume13-
dc.identifier.issue1-
dc.identifier.spage173-
dc.identifier.epage204-
dc.identifier.eissn1540-3467-

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