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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

An adaptive stochastic Galerkin method for random elliptic operators


Author: Claude Jeffrey Gittelson
Journal: Math. Comp. 82 (2013), 1515-1541
MSC (2010): Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20
Published electronically: February 12, 2013
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Abstract: We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence of the random solution, which combines with any discretization on the spatial domain. In addition to selecting active polynomial modes, this solver can adaptively construct a separate spatial discretization for each of their coefficients. We show convergence of the solver in this general setting, along with a computable bound for the mean square error, and an optimality property in the case of a single spatial discretization. Numerical computations demonstrate convergence of the solver and compare it to a sparse tensor product construction.


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Additional Information

Claude Jeffrey Gittelson
Affiliation: Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zurich, Switzerland
Address at time of publication: Department of Mathematics, Purdue University, 150 N.\ University Street, West Lafayette, Indiana 47907
Email: cgittels@purdue.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02654-3
PII: S 0025-5718(2013)02654-3
Received by editor(s): March 2, 2011
Received by editor(s) in revised form: September 24, 2011
Published electronically: February 12, 2013
Additional Notes: This research was supported in part by the Swiss National Science Foundation grant No. 200021-120290/1.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.