Sparse second moment analysis for elliptic problems in stochastic domains

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 work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain. 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.