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Sparse second moment analysis for elliptic problems in stochastic domains

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We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially \({\mathcal{O}(N)}\) work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain.

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Correspondence to Helmut Harbrecht.

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This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.

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Harbrecht, H., Schneider, R. & Schwab, C. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109, 385–414 (2008).

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