Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions

Haibing Wang Southeast University Yi Li Southeast University

Numerical Analysis and Scientific Computing mathscidoc:2103.25007

Journal of Computational Physics, 369, 1-15, 2018.5
We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.
Inverse problem, Unknown inclusions, Heat equation, Layer potential
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@inproceedings{haibing2018numerical,
  title={Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions},
  author={Haibing Wang, and Yi Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210325173524644098766},
  booktitle={Journal of Computational Physics},
  volume={369},
  pages={1-15},
  year={2018},
}
Haibing Wang, and Yi Li. Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions. 2018. Vol. 369. In Journal of Computational Physics. pp.1-15. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210325173524644098766.
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