Expansion of random boundary excitations for elliptic PDEs

Sabelfeld Karl

Monte Carlo Methods and Applications, 2008, vol. 13, issue 5-6, 405-453

Abstract: In this paper we deal with elliptic boundary value problems with random boundary conditions. Solutions to these problems are inhomogeneous random fields which can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen–Loève series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous random fields. We study the correlation structure of solutions to some classical elliptic equations in respond to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to Laplace equation, biharmonic equation, and to the Lamé system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an excellent benchmark for developing numerical methods, e.g., Monte Carlo simulations, stochastic volume and boundary element methods.

Keywords: White noise; generalized random processes; Karhunen–Loève expansion; Poisson integral formula; random boundary excitations; Laplace; biharmonic; and Lamé equations (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1515/mcma.2007.022

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