Numerical Realization Of Probe Method In Inverse Scattering Problem

Inverse scatterings belong to the category of inverse problems, which arise in many application areas such as nondestructive test, medical imaging and geophysics. The tasks of these problems are to identify the physics property of scatterers from measurement of scattered wave field. Mathematically, they are described by coefficient (boundary) inversions of elliptic equations. Probe method is one of the most important inversion schemes developed recently for inverse scattering problems. Its main idea is to construct an indicator function with parameter point outside scatterer from information about scattered wave such as far-field pattern. When the parameter point approaches to the boundary of scatterer, the indicator function blows up. Thus the boundary is reconstructed from the indicator behavior. The key to probe method is to construct a Dirichlet-to-Neumann(D-to-N) map defined on the boundary of a known domain Ω which contains the scatterer, and then study the indicator behavior construced from D-to-N map. The purpose of this thesis is to study the numerical realization of probe method for a scatterer with sound-soft boundary, and test the realization performance of probe method from the simulation data. Some numerical phoenomena as well as the possible solving methods are considered. In order to simplify the test procedure, we consider the numerical realization from D-to-N map directly. There are three parts in the paper. First of all, by brifely introducing the potential method for solving Helmholtz equation in a non-conneced domain, the simulation of D-to-N map is generated for our inverse problem. Since the construction and numerical performance of Runge approximation is an important step in probe method, the second work in this thesis is to discuss the construction of Runge approximation by an optimization technique for an integral equation of first kind. Finally, the numerical realization of probe method is considered based on the Runge approximation scheme proposed in this paper.