Numerical Realization Of Probe Method For Multiple Obstacles

Inverse scattering problems for acoustic wave are very important in the applied science and numerical analysis. They aim to reconstruct the shape of the obstacle from the information of scattered wave such as its far-field pattern. The optimization technique has been applied widely for the reconstruction of obstacle boundary, but with the restriction of known obstacle types. This paper considers the reconstruction of two obstacles with different boundary type by probe method, including recovery of the boundary shape and boundary type simultaneously. The basic idea of this method is to construct an indicator function which blows up in different ways on the boundaries for obstacles with different boundary types. Our paper concerns with the numerical realization of this method for multiple obstacles. By constructing the Runge approximation function and the cone-shape domain suitably, this paper firstly gives a numerical test of probe method for multiple obstacles and obtains some conclusions.Our paper considers three parts.Firstly, we consider the choice of needle for probe method and the construction of approximated domain in the case of multiple obstacles. By solving an integral equation of the first kind with the minimum norm solution, we construct the Runge approximation function successfully. These data are used in our construction of indicator function.Secondly, we consider the simulation of Neumann data in the indicator function by solving the direct problem. Theoretically, these data should be solved from the far-field pattern from an integral equation with hyper-singularity. To avoid the error in this complicated procedure, we generate these data by simulation procedure, which is obtained from boundary integral equation method. This procedure lessens the amount of computation and avoids the differential operation in obtaining the Neumann data.Finally, we realize the probe method by giving some numerical performances for different scattering configurations. Our results indicate that we can not expect too much for the accuracy of this method, provided we assume the blowing up criterion uniformly for different points of the boundary. However, the approximate shape obtained from this inversion algorithm can be used as the initial guess for other iteration-based reconstruction schemes.