A simple method for solving the inverse scattering problem for the difference Helmholtz equation

The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We use the finite-difference approximation for the Helmholtz equation along with the exact radiation condition for the discrete equation. The approach involves two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new approach.