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A fast deterministic method for stochastic elliptic interface problems based on low-rank approximation

by H. Harbrecht and J. Li

(Report number 2011-24)

Abstract
In this work, we propose a fast deterministic numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-type Taylor expansion in the framework of the asymptotic perturbation approach. Given a priori known mean field and two-point correlation function of random interface variations, we can quantify the mean field and variance of random solutions in terms of certain orders of the perturbation magnitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for interface-resolved finite element approximation in both physical and stochastic dimensions. In particular, a fast finite difference scheme is proposed to compute the two-point correlation function of random solutions using the low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to demonstrate the advantages of the proposed method.

Keywords:

@Techreport{HL11_140,
  author = {H. Harbrecht and J. Li},
  title = {A fast deterministic method for stochastic elliptic interface problems based on low-rank approximation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-24.pdf },
  year = {2011}
}

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