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Article: A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms

TitleA dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms
Authors
KeywordsUncertainty quantification
Bi-orthogonality
Karhunen-Loeve expansion
Reduced-order model
Stochastic partial differential equations
Issue Date2013
Citation
Journal of Computational Physics, 2013, v. 242, p. 843-868 How to Cite?
AbstractWe propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen-Loeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [11], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method. © 2013 Elsevier Inc.
Persistent Identifierhttp://hdl.handle.net/10722/219698
ISSN
2015 Impact Factor: 2.556
2015 SCImago Journal Rankings: 2.167

 

DC FieldValueLanguage
dc.contributor.authorCheng, Mulin-
dc.contributor.authorHou, Thomas Y.-
dc.contributor.authorZhang, Zhiwen-
dc.date.accessioned2015-09-23T02:57:45Z-
dc.date.available2015-09-23T02:57:45Z-
dc.date.issued2013-
dc.identifier.citationJournal of Computational Physics, 2013, v. 242, p. 843-868-
dc.identifier.issn0021-9991-
dc.identifier.urihttp://hdl.handle.net/10722/219698-
dc.description.abstractWe propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen-Loeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [11], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method. © 2013 Elsevier Inc.-
dc.languageeng-
dc.relation.ispartofJournal of Computational Physics-
dc.subjectUncertainty quantification-
dc.subjectBi-orthogonality-
dc.subjectKarhunen-Loeve expansion-
dc.subjectReduced-order model-
dc.subjectStochastic partial differential equations-
dc.titleA dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms-
dc.typeArticle-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.doi10.1016/j.jcp.2013.02.033-
dc.identifier.scopuseid_2-s2.0-84875865117-
dc.identifier.volume242-
dc.identifier.spage843-
dc.identifier.epage868-
dc.identifier.eissn1090-2716-

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