In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method.
Mots-clés : elliptic interface problem, stochastic interface, low-rank approximation, pivoted Cholesky decomposition
@article{M2AN_2013__47_5_1533_0,
author = {Harbrecht, Helmut and Li, Jingzhi},
title = {First order second moment analysis for stochastic interface problems based on low-rank approximation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1533--1552},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013079},
mrnumber = {3100774},
zbl = {1297.65009},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013079/}
}
TY - JOUR AU - Harbrecht, Helmut AU - Li, Jingzhi TI - First order second moment analysis for stochastic interface problems based on low-rank approximation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1533 EP - 1552 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013079/ DO - 10.1051/m2an/2013079 LA - en ID - M2AN_2013__47_5_1533_0 ER -
%0 Journal Article %A Harbrecht, Helmut %A Li, Jingzhi %T First order second moment analysis for stochastic interface problems based on low-rank approximation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1533-1552 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013079/ %R 10.1051/m2an/2013079 %G en %F M2AN_2013__47_5_1533_0
Harbrecht, Helmut; Li, Jingzhi. First order second moment analysis for stochastic interface problems based on low-rank approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1533-1552. doi : 10.1051/m2an/2013079. http://www.numdam.org/articles/10.1051/m2an/2013079/
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