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Article: An adaptive ANOVA-based data-driven stochastic method for elliptic PDEs with random coefficient

TitleAn adaptive ANOVA-based data-driven stochastic method for elliptic PDEs with random coefficient
Authors
KeywordsAnalysis of variance
Uncertainty quantification
Data-driven methods
Karhunen-Loeve expansion
Model reduction
Stochastic partial differential equations
Issue Date2014
Citation
Communications in Computational Physics, 2014, v. 16, n. 3, p. 571-598 How to Cite?
AbstractIn this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of lowdimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one-and two-dimensional elliptic PDEs with random coefficients. © 2014 Global-Science Press.
Persistent Identifierhttp://hdl.handle.net/10722/219752
ISSN
2015 Impact Factor: 1.778
2015 SCImago Journal Rankings: 1.198

 

DC FieldValueLanguage
dc.contributor.authorZhang, Zhiwen-
dc.contributor.authorHu, Xin-
dc.contributor.authorHou, Thomas Y.-
dc.contributor.authorLin, Guang-
dc.contributor.authorYan, Mike-
dc.date.accessioned2015-09-23T02:57:53Z-
dc.date.available2015-09-23T02:57:53Z-
dc.date.issued2014-
dc.identifier.citationCommunications in Computational Physics, 2014, v. 16, n. 3, p. 571-598-
dc.identifier.issn1815-2406-
dc.identifier.urihttp://hdl.handle.net/10722/219752-
dc.description.abstractIn this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of lowdimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one-and two-dimensional elliptic PDEs with random coefficients. © 2014 Global-Science Press.-
dc.languageeng-
dc.relation.ispartofCommunications in Computational Physics-
dc.subjectAnalysis of variance-
dc.subjectUncertainty quantification-
dc.subjectData-driven methods-
dc.subjectKarhunen-Loeve expansion-
dc.subjectModel reduction-
dc.subjectStochastic partial differential equations-
dc.titleAn adaptive ANOVA-based data-driven stochastic method for elliptic PDEs with random coefficient-
dc.typeArticle-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.doi10.4208/cicp.270913.020414a-
dc.identifier.scopuseid_2-s2.0-84904422581-
dc.identifier.volume16-
dc.identifier.issue3-
dc.identifier.spage571-
dc.identifier.epage598-
dc.identifier.eissn1991-7120-

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