On the reconstruction of obstacle using MFS from the far field data

We consider the reconstruction of an obstacle with impedance boundary in an inverse scattering problem from far-field data. Both the boundary shape and the boundary impedance are unknown. By first transforming the far field into the near field, we consider this inverse problem as the detection of a void with impedance boundary embedded in bounded homogeneous media domain G. Then we set the Dirichlet-to-Neumann boundary condition in G and solve this interior boundary reconstruction problem in terms of the method of fundamental solution (MFS) with the near field on G as input inversion data. For this problem, both the locations of source points and the obstacle boundary as well as the expansion coefficients of wave field are taken to be the unknowns under the framework of regularizing optimization. The property of this optimization problem such as the existence and convergence of the minimizer is analysed. Numerical examples are presented for several configurations to show the validity of our proposed scheme.