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IMA Journal of Applied Mathematics Advance Access published online on December 14, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm057
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A procedure for the reconstruction of a stochastic stationary temperature field

B. Tomas Johansson{dagger}

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

{dagger} Email: b.t.johansson{at}bham.ac.uk

Received on June 4, 2007; Accepted on October 8, 2007

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero

Keywords: finite element; ill posed; Karhunen–Loève expansion; stochastic elliptic equation.


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