A variational method for identifying a spacewise-dependent heat source

B. Tomas Johansson, Daniel Lesnic

Research output: Contribution to journalArticle

Abstract

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
Volume72
Issue number6
Early online date3 Sep 2007
DOIs
Publication statusPublished - 3 Sep 2007

Fingerprint

Heat Source
Variational Methods
Temperature Measurement
Iteration
Heat Equation
Parabolic Equation
Dependent
Inverse Problem
Inverse problems
Temperature measurement
Discrepancy Principle
Numerical Results
Random Noise
Conjugate Gradient
Iterative Procedure
Numerical Approximation
Instant
Unique Solution
Iterative Algorithm
Boundary Elements

Keywords

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

Cite this

Johansson, B. Tomas ; Lesnic, Daniel. / A variational method for identifying a spacewise-dependent heat source. In: IMA Journal of Applied Mathematics. 2007 ; Vol. 72, No. 6. pp. 748-760.
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A variational method for identifying a spacewise-dependent heat source. / Johansson, B. Tomas; Lesnic, Daniel.

In: IMA Journal of Applied Mathematics, Vol. 72, No. 6, 03.09.2007, p. 748-760.

Research output: Contribution to journalArticle

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AU - Lesnic, Daniel

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