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Mathematics of Computation

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First order $ k$-th moment finite element analysis of nonlinear operator equations with stochastic data

Authors: Alexey Chernov and Christoph Schwab
Journal: Math. Comp. 82 (2013), 1859-1888
MSC (2010): Primary 65N30, 65J15
Published electronically: April 4, 2013
MathSciNet review: 3073184
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Abstract: We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations $ J(\alpha ,u)=0$ for random input $ \alpha (\omega )$ with almost sure realizations in a neighborhood of a nominal input parameter $ \alpha _0$. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, $ u(\omega ) = S(\alpha (\omega ))$.

We derive a multilinear, tensorized operator equation for the deterministic computation of $ k$-th order statistical moments of the random solution's fluctuations $ u(\omega ) - S(\alpha _0)$. We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the $ k$-th statistical moment equation. We prove a shift theorem for the $ k$-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

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Additional Information

Alexey Chernov
Affiliation: Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, 53115 Bonn, Germany

Christoph Schwab
Affiliation: Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Keywords: Nonlinear operator equations, random parameters, deterministic methods, Fr\'echet derivative, sparse tensor approximation, random domain
Received by editor(s): August 12, 2011
Received by editor(s) in revised form: February 5, 2012
Published electronically: April 4, 2013
Additional Notes: This work was supported in part under grant ERC AdG 247277 (to CS), initiated at the Institute for Mathematics and Applications (IMA) in Minnesota, USA and completed at the Hausdorff Research Institute for Mathematics, Bonn, Germany during the trimester “High dimensional problems” May–August 2011. The first author acknowledges support by the Hausdorff Center for Mathematics, Bonn and the second author acknowledges the excellent working conditions at the Hausdorff Research Institute
Dedicated: Dedicated to W. L. Wendland on the occasion of his 75th anniversary
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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