## Abstract

We investigate the problem of determining the stationary temperature

field on an inclusion from given Cauchy data on an accessible exterior

boundary. On this accessible part the temperature (or the heat flux) is known,

and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in

combination with Tikhonov regularization for the stable determination of the

temperature and flux on the inclusion. To determine these quantities on the

inclusion, boundary integral equations are derived using Green’s functions, and

properties of these equations are shown in an L2-setting. An effective way of

discretizing these boundary integral equations based on the Nystr¨om method

and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations

can be obtained with small computational effort, and the accuracy is increasing

with the length of the portion of the boundary where the additionally data is

given.

field on an inclusion from given Cauchy data on an accessible exterior

boundary. On this accessible part the temperature (or the heat flux) is known,

and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in

combination with Tikhonov regularization for the stable determination of the

temperature and flux on the inclusion. To determine these quantities on the

inclusion, boundary integral equations are derived using Green’s functions, and

properties of these equations are shown in an L2-setting. An effective way of

discretizing these boundary integral equations based on the Nystr¨om method

and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations

can be obtained with small computational effort, and the accuracy is increasing

with the length of the portion of the boundary where the additionally data is

given.

Original language | English |
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Pages (from-to) | 25–38 |

Number of pages | 13 |

Journal | Inverse Problems and Imaging |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2012 |

## Keywords

- cauchy problem
- integral equation of the first kind
- logarithmic- and hypersingularities
- single- and double layer potentials
- Green’s function
- Laplace equation
- Tikhonov regularization