GCRIS

Our research is motivated by the classical inverse scattering problem to reconstruct impedance functions. This problem is ill-posed and nonlinear. This problem can be solved by Newton-type iterative and regularization methods. In the first part, we suggest numerical methods for resolving the generalized impedance boundary value problem for the modified Helmholtz equation. We follow some strategies to solve it. The strategies of the first method are founded on the idea that the problem can be reduced to the boundary integral equation with a hyper-singular kernel. While the strategy of the second approach makes use of the concept of numerical differentiation, the first approach treats the hyper singular integral operator by splitting off the singularity. We also show the convergence of the first method in the Sobolev sense and the solvability of the boundary integral equation. We give numerical examples which show exponential convergence for analytical data. In the second part of this work, we take into account the inverse scattering problem of reconstructing the cavity’s surface impedance from sources and measurements positioned on a curve within it. For the approximate solution of an ill-posed and nonlinear problem, we propose a direct and hybrid method which is a Newton-type method based on a boundary integral equation approach for the boundary value problem for the modified Helmholtz equation. As a consequence of this, the numerical algorithm combines the benefits of direct and iterative schemes and has the same level of accuracy as a Newton-type method while not requiring an initial guess. The results are confirmed by numerical examples which show that the numerical method is feasible and effective.

Phd Degree / Doktora

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