A variational method for identifying a spacewise-dependent heat source

Bjorn Johansson, D Lesnic

Research output: Contribution to journalArticle

91 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)748-760
Number of pages13
JournalIMA Journal of Applied Mathematics
Issue number6
Publication statusPublished - 16 Aug 2007


  • heat source
  • parabolic heat equation
  • boundary element method
  • iterative regularization
  • inverse problem
  • conjugate gradient method
  • discrepancy principle


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