A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions

Handle URI:
http://hdl.handle.net/10754/597379
Title:
A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
Authors:
Butler, T.; Dawson, C.; Wildey, T.
Abstract:
We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.
Citation:
Butler T, Dawson C, Wildey T (2011) A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions. SIAM Journal on Scientific Computing 33: 1267–1291. Available: http://dx.doi.org/10.1137/100795760.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
Issue Date:
Jan-2011
DOI:
10.1137/100795760
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
Submitted to the journal's Methods and Algorithms for Scientific Computing section May 18, 2010; accepted for publication (in revised form) March 2, 2011; published electronically June 7, 2011. This work was made possible with funding from the King Abdullah University of Science and Technology (KAUST).Sandia National Labs, Albuquerque, NM 87185 (tmwilde@sandia.gov). Sandia is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94-AL85000. This author's work was partially supported by NSF grant DMS 0618679.
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Full metadata record

DC FieldValue Language
dc.contributor.authorButler, T.en
dc.contributor.authorDawson, C.en
dc.contributor.authorWildey, T.en
dc.date.accessioned2016-02-25T12:32:00Zen
dc.date.available2016-02-25T12:32:00Zen
dc.date.issued2011-01en
dc.identifier.citationButler T, Dawson C, Wildey T (2011) A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions. SIAM Journal on Scientific Computing 33: 1267–1291. Available: http://dx.doi.org/10.1137/100795760.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/100795760en
dc.identifier.urihttp://hdl.handle.net/10754/597379en
dc.description.abstractWe develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipSubmitted to the journal's Methods and Algorithms for Scientific Computing section May 18, 2010; accepted for publication (in revised form) March 2, 2011; published electronically June 7, 2011. This work was made possible with funding from the King Abdullah University of Science and Technology (KAUST).Sandia National Labs, Albuquerque, NM 87185 (tmwilde@sandia.gov). Sandia is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94-AL85000. This author's work was partially supported by NSF grant DMS 0618679.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectA posteriori error analysisen
dc.subjectAdjoint problemen
dc.subjectPolynomial chaosen
dc.subjectStochastic spectral methodsen
dc.titleA Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansionsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionUniversity of Texas at Austin, Austin, United Statesen
dc.contributor.institutionSandia National Laboratories, New Mexico, Albuquerque, United Statesen
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