Abstract
We consider inverse boundary value problems for the Helmholtz and Laplace equations in domains with corners. This thesis has three main parts. The first one gives the functional analytic concepts. In the second part we consider the two-dimensional exterior Dirichlet problem for the Helmholtz equation. The inverse problem consists of determining the nonsmooth boundary from knowledge of the far field pattern for the scattering of time harmonic plane waves. Frechet differentiability with respect to the boundary is shown for the far field operator, which for a fixed incident wave maps the boundary onto the far field pattern of the scattered wave. For the approximate solution of this ill-posed and nonlinear problem we present a regularized Newton iteration scheme. For the sake of completeness, first we answer the questions about uniqueness and existence of the corresponding direct problem via an integral equation method, then we solve this problem numerically. In the third part we consider a direct and inverse two-dimensional problem in impedance tomography for the Laplace equation. The direct problem consists of determining the current on a section that is not corroded. For the inverse problem we know the Dirichlet to Neumann map and we have to determine the corroded boundary. We solve the inverse problem again with a Newton method
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