On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods

Handle URI:
http://hdl.handle.net/10754/562295
Title:
On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods
Authors:
Beck, Joakim; Tempone, Raul ( 0000-0003-1967-4446 ) ; Nobile, Fabio; Tamellini, Lorenzo
Abstract:
In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
KAUST Department:
Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
Issue Date:
Sep-2012
DOI:
10.1142/S0218202512500236
Type:
Article
ISSN:
02182025
Sponsors:
The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project No. 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and uncertainty quantification for models of porous media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project No. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".
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.authorBeck, Joakimen
dc.contributor.authorTempone, Raulen
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTamellini, Lorenzoen
dc.date.accessioned2015-08-03T09:59:44Zen
dc.date.available2015-08-03T09:59:44Zen
dc.date.issued2012-09en
dc.identifier.issn02182025en
dc.identifier.doi10.1142/S0218202512500236en
dc.identifier.urihttp://hdl.handle.net/10754/562295en
dc.description.abstractIn this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.en
dc.description.sponsorshipThe authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project No. 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and uncertainty quantification for models of porous media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project No. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectbest M-terms polynomial approximationen
dc.subjectelliptic equationsen
dc.subjectmultivariate polynomial approximationen
dc.subjectPDEs with random dataen
dc.subjectSmolyak approximationen
dc.subjectsparse gridsen
dc.subjectstochastic collocation methodsen
dc.subjectstochastic Galerkin methodsen
dc.subjectUncertainty quantificationen
dc.titleOn the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methodsen
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.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionMOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 22-20133 Milano, Italyen
dc.contributor.institutionCSQI, MATHICSE, Ecole Politechnique Fédérale de Lausanne, CH 1015, Lausanne, Switzerlanden
kaust.authorTempone, Raulen
kaust.authorBeck, Joakimen
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