Multi-Index Stochastic Collocation for random PDEs

Handle URI:
http://hdl.handle.net/10754/603944
Title:
Multi-Index Stochastic Collocation for random PDEs
Authors:
Haji Ali, Abdul Lateef ( 0000-0002-6243-0335 ) ; Nobile, Fabio; Tamellini, Lorenzo; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Multi-Index Stochastic Collocation for random PDEs 2016 Computer Methods in Applied Mechanics and Engineering
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
28-Mar-2016
DOI:
10.1016/j.cma.2016.03.029
Type:
Article
ISSN:
00457825
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0045782516301141
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHaji Ali, Abdul Lateefen
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTamellini, Lorenzoen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2016-03-30T07:26:10Zen
dc.date.available2016-03-30T07:26:10Zen
dc.date.issued2016-03-28en
dc.identifier.citationMulti-Index Stochastic Collocation for random PDEs 2016 Computer Methods in Applied Mechanics and Engineeringen
dc.identifier.issn00457825en
dc.identifier.doi10.1016/j.cma.2016.03.029en
dc.identifier.urihttp://hdl.handle.net/10754/603944en
dc.description.abstractIn this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0045782516301141en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 28 March 2016. DOI: 10.1016/j.cma.2016.03.029en
dc.subjectUncertainty Quantificationen
dc.subjectRandom PDEsen
dc.subjectMultivariate approximationen
dc.subjectSparse gridsen
dc.subjectStochastic Collocation methodsen
dc.subjectMultilevel methodsen
dc.subjectCombination techniqueen
dc.titleMulti-Index Stochastic Collocation for random PDEsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.eprint.versionPost-printen
dc.contributor.institutionCSQI - MATHICSE, École Polytechnique Fédérale de Lausanne, Station 8, CH 1015, Lausanne, Switzerlanden
dc.contributor.institutionDipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italyen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorHaji Ali, Abdul Lateefen
kaust.authorTempone, Raulen
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