First order second moment analysis for stochastic interface problems based on low-rank approximation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, p. 1533-1552

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.

DOI : https://doi.org/10.1051/m2an/2013079
Classification:  60H15,  60H35,  65C20,  65C30
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     pages = {1533-1552},
     doi = {10.1051/m2an/2013079},
     zbl = {1297.65009},
     mrnumber = {3100774},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, pp. 1533-1552. doi : 10.1051/m2an/2013079. http://www.numdam.org/item/M2AN_2013__47_5_1533_0/

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