Numerische Mathematik

, Volume 109, Issue 3, pp 385–414

Sparse second moment analysis for elliptic problems in stochastic domains

  • Helmut Harbrecht
  • Reinhold Schneider
  • Christoph Schwab


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.

Mathematics Subject Classification (2000)

35J20 35R60 65N38 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Helmut Harbrecht
    • 1
  • Reinhold Schneider
    • 2
  • Christoph Schwab
    • 3
  1. 1.Institut für Numerische SimulationUniversität BonnBonnGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Seminar für Angewandte MathematikEidgenössische Technische Hochschule ZürichZürichSwitzerland

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