Sparse second moment analysis for elliptic problems in stochastic domains

Authors:
Helmut Harbrecht;Reinhold Schneider;Christoph Schwab
Affiliations:
Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115, Bonn, Germany;Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623, Berlin, Germany;Eidgenössische Technische Hochschule Zürich, Seminar für Angewandte Mathematik, Rämistr. 101, 8092, Zürich, Switzerland
Venue:
Numerische Mathematik
Year:
2008

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Abstract

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.