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An inverse boundary value problem for the heat equation: The Neumann condition. (English) Zbl 1044.35527

Summary: We consider the inverse problem to determine the shape of an insulated inclusion within a heat conducting medium from overdetermined Cauchy data of solutions for the heat equation on the accessible exterior boundary of the medium. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton iteration scheme based on a boundary integral equation approach for the initial Neumann boundary value problem for the heat equation. For a foundation of the Newton method we establish the differentiability of the solution to the initial Neumann boundary value problem with respect to the interior boundary curve in the sense of a domain derivative and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented.

MSC:

35R30 Inverse problems for PDEs
35K05 Heat equation
80A22 Stefan problems, phase changes, etc.
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