A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

Handle URI:
http://hdl.handle.net/10754/555664
Title:
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
Authors:
Babuška, Ivo; Nobile, Fabio; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2010, 52 (2):317 SIAM Review
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Review
Issue Date:
Jan-2010
DOI:
10.1137/100786356
Type:
Article
ISSN:
0036-1445; 1095-7200
Additional Links:
http://epubs.siam.org/doi/abs/10.1137/100786356
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorBabuška, Ivoen
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-05-25T08:27:41Zen
dc.date.available2015-05-25T08:27:41Zen
dc.date.issued2010-01en
dc.identifier.citationA Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2010, 52 (2):317 SIAM Reviewen
dc.identifier.issn0036-1445en
dc.identifier.issn1095-7200en
dc.identifier.doi10.1137/100786356en
dc.identifier.urihttp://hdl.handle.net/10754/555664en
dc.description.abstractThis work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/100786356en
dc.rightsArchived with thanks to SIAM Reviewen
dc.subjectstochastic collocation methoden
dc.subjectpartial differential equations with random inputsen
dc.subjectfinite elementsen
dc.subjectuncertainty quantificationen
dc.subjectconvergence ratesen
dc.subjectmultivariate polynomial approximationen
dc.subjectSmolyak approximationen
dc.subjectanisotropic sparse approximationen
dc.titleA Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Dataen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalSIAM Reviewen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionICES, The University of Texas at Austin, Austin, TX 78712en
dc.contributor.institutionMOX, Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italyen
kaust.authorTempone, Raulen
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