In this paper, we consider the inverse scattering problem of recovering the shape of a perfectly conducting cavity from one source and several measurements placed on a curve inside the cavity. Under restrictive assumptions on the size of the cavity, a uniqueness theorem for finitely many excitations is given. Based on a system of nonlinear and ill-posed integral equations for the unknown boundary, we apply a regularized Newton iterative approach to find the boundary. We present the mathematical foundation of the method and give several numerical examples to show the viability of the method.
We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ∂ D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ∂ D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.
a b s t r a c tIn this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green's formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.
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