Integral Equation Methods in Inverse Obstacle Scattering with a Generalized Impedance Boundary Condition
The inverse problem under consideration is to reconstruct the shape of an impenetrable two-dimensional obstacle with a generalized impedance boundary condition from the far field pattern for scattering of time-harmonic acoustic or E-polarized electromagnetic plane waves. We propose an inverse algorithm that extends the approach suggested by Johansson and Sleeman (IMA J. Appl. Math. 72(1):96–112, 2007) for the case of the inverse problem for a sound-soft or perfectly conducting scatterer. It is based on a system of nonlinear boundary integral equations associated with a single-layer potential approach to solve the forward scattering problem which extends the integral equation method proposed by Cakoni and Kress (Inverse Prob. 29(1):015005, 2013) for a related boundary value problem for the Laplace equation. In addition, we also present an algorithm for reconstructing the impedance function when the shape of the scatterer is known. We present the mathematical foundations of the methods and exhibit their feasibility by numerical examples.
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