Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

Handle URI:
http://hdl.handle.net/10754/561427
Title:
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
Authors:
Nobile, Fabio; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
We consider the problem of numerically approximating statistical moments of the solution of a time- dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen-Loève expansions driven by a finite number of uncorrelated random variables. After approxi- mating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. © 2009 John Wiley & Sons, Ltd.
KAUST Department:
Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
Wiley-Blackwell
Journal:
International Journal for Numerical Methods in Engineering
Issue Date:
5-Nov-2009
DOI:
10.1002/nme.2656
Type:
Article
ISSN:
00295981
Sponsors:
The first and second authors were partially supported by the University of Austin Subcontract (Project Number 024550. Center for Predictive Computational Science) The first author acknowledges the Italian fund PRIN 2007 Numerical modeling, For scientific computing and advanced applications'. The Second acknowledges his Dahlquist fellowship at the Royal Institute of Technology in Stockholm. Sweden and his Start up funds at SC. Florida State University. He also would like to acknowledge the support of UdelaR in Uruguay.
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-02T09:11:01Zen
dc.date.available2015-08-02T09:11:01Zen
dc.date.issued2009-11-05en
dc.identifier.issn00295981en
dc.identifier.doi10.1002/nme.2656en
dc.identifier.urihttp://hdl.handle.net/10754/561427en
dc.description.abstractWe consider the problem of numerically approximating statistical moments of the solution of a time- dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen-Loève expansions driven by a finite number of uncorrelated random variables. After approxi- mating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. © 2009 John Wiley & Sons, Ltd.en
dc.description.sponsorshipThe first and second authors were partially supported by the University of Austin Subcontract (Project Number 024550. Center for Predictive Computational Science) The first author acknowledges the Italian fund PRIN 2007 Numerical modeling, For scientific computing and advanced applications'. The Second acknowledges his Dahlquist fellowship at the Royal Institute of Technology in Stockholm. Sweden and his Start up funds at SC. Florida State University. He also would like to acknowledge the support of UdelaR in Uruguay.en
dc.publisherWiley-Blackwellen
dc.subjectMonte carlo samplingen
dc.subjectMultivariate polynomial approximationen
dc.subjectParabolic equationsen
dc.subjectPDEs with random dataen
dc.subjectPoint Collocationen
dc.subjectSmolyak approximationen
dc.subjectSparse gridsen
dc.subjectStochastic Collocation methodsen
dc.subjectStochastic galerkin methodsen
dc.titleAnalysis and implementation issues for the numerical approximation of parabolic equations with random coefficientsen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalInternational Journal for Numerical Methods in Engineeringen
dc.contributor.institutionMOX, Dipartimento di Matematica, Politecnico di Milano, Italyen
dc.contributor.institutionSchool of Computer Sciences and Communication, KTH, S-100 44 Stockholm, Swedenen
kaust.authorTempone, Raulen
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