### Abstract

Original language | English |
---|---|

Pages (from-to) | 748-760 |

Number of pages | 13 |

Journal | IMA Journal of Applied Mathematics |

Volume | 72 |

Issue number | 6 |

Early online date | 3 Sep 2007 |

DOIs | |

Publication status | Published - 3 Sep 2007 |

### Fingerprint

### Keywords

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

### Cite this

*IMA Journal of Applied Mathematics*,

*72*(6), 748-760. https://doi.org/10.1093/imamat/hxm024

}

*IMA Journal of Applied Mathematics*, vol. 72, no. 6, pp. 748-760. https://doi.org/10.1093/imamat/hxm024

**A variational method for identifying a spacewise-dependent heat source.** / Johansson, B. Tomas; Lesnic, Daniel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A variational method for identifying a spacewise-dependent heat source

AU - Johansson, B. Tomas

AU - Lesnic, Daniel

PY - 2007/9/3

Y1 - 2007/9/3

N2 - 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.

AB - 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.

KW - boundary element method

KW - conjugate gradient method

KW - discrepancy principle

KW - heat source

KW - inverse problem

KW - iterative regularization

KW - parabolic heat equation

UR - http://imamat.oxfordjournals.org/content/72/6/748

U2 - 10.1093/imamat/hxm024

DO - 10.1093/imamat/hxm024

M3 - Article

VL - 72

SP - 748

EP - 760

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 6

ER -