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**On the numerical solution of an inverse boundary value problem for the heat equation.**
*(English)*
Zbl 0917.35157

Regularized Newton methods for the solution of inverse obstacle scattering problems for time-harmonic waves, i.e. for inverse boundary value problems for the Helmholtz equation have recently been studied and successfully applied by various authors.

In this paper, the authors consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem they propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of this method.

The paper is organized as follows. In section 2 the differentiability of the solution to the initial boundary value problem with respect to the interior boundary is established. This is done within the framework of the domain derivative in a weak solution setting and serves as the theoretical foundation of the Newton method for the approximate solution considered in the second part of the paper. Since each step of a Newton scheme will require the numerical solution of the direct problem, in section 3 the approximate solution for a given interior curve is described via boundary integral equations of the first kind. In section 4 the authors develop the Newton scheme, i.e. they describe the updating of the interior curve by one step of the Newton iteration. Finally, section 5 is devoted to numerical examples for the solution of the inverse problem.

In this paper, the authors consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem they propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of this method.

The paper is organized as follows. In section 2 the differentiability of the solution to the initial boundary value problem with respect to the interior boundary is established. This is done within the framework of the domain derivative in a weak solution setting and serves as the theoretical foundation of the Newton method for the approximate solution considered in the second part of the paper. Since each step of a Newton scheme will require the numerical solution of the direct problem, in section 3 the approximate solution for a given interior curve is described via boundary integral equations of the first kind. In section 4 the authors develop the Newton scheme, i.e. they describe the updating of the interior curve by one step of the Newton iteration. Finally, section 5 is devoted to numerical examples for the solution of the inverse problem.

Reviewer: Gabriel Dimitriu (Iaşi)

### MSC:

35R30 | Inverse problems for PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

49M15 | Newton-type methods |