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IMA Journal of Applied Mathematics Advance Access originally published online on September 3, 2007
IMA Journal of Applied Mathematics 2007 72(6):748-760; doi:10.1093/imamat/hxm024
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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 variational method for identifying a spacewise-dependent heat source

B. Tomas Johansson

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

Daniel Lesnic

Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received on October 23, 2006; Accepted on March 30, 2007

The inverse problem of determining a spacewise-dependent heat source for the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time is studied. This spacewise-dependent temperature measurement ensures that this inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. We propose a variational conjugate gradient-type iterative algorithm for the stable reconstruction of the heat source based on a sequence of well-posed direct problems for the parabolic heat equation which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterative procedure at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented which have the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure yields stable and accurate numerical approximations after only a few iterations.

Keywords: boundary element method; conjugate gradient method; discrepancy principle; heat source; inverse problem; iterative regularization; parabolic heat equation.


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