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First order second moment analysis for stochastic interfaceproblems based on low-rank approximation∗

Published online by Cambridge University Press:  14 August 2013

Helmut Harbrecht  and
Jingzhi Li
Helmut Harbrecht
Affiliation:
Helmut Harbrecht, Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland.. helmut.harbrecht@unibas.ch
Jingzhi Li
Affiliation:
Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, P. R. China.; li.jz@sustc.edu.cn
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Abstract

In this paper, we propose a numerical method to solve stochastic elliptic interfaceproblems with random interfaces. Shape calculus is first employed to derive theshape-Taylor expansion in the framework of the asymptotic perturbation approach. Given themean field and the two-point correlation function of the random interface, we can thusquantify the mean field and the variance of the random solution in terms of certain ordersof the perturbation amplitude by solving a deterministic elliptic interface problem andits tensorized counterpart with respect to the reference interface. Error estimates arederived for the interface-resolved finite element approximation in both, the physical andthe stochastic dimension. In particular, a fast finite difference scheme is proposed tocompute the variance of random solutions by using a low-rank approximation based on thepivoted Cholesky decomposition. Numerical experiments are presented to validate andquantify the method.

Keywords

Elliptic interface problemstochastic interfacelow-rank approximationpivoted Cholesky decomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1533 - 1552
Copyright
© EDP Sciences, SMAI, 2013

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