The Two Dimensional Inverse Scattering Problem Of Acoustic Wave

Scattering theory has played a central role in twentieth century mathematic-physics, The inverse scattering problem of acoustic wave is a typical mathematic physics inverse problem, this paper introduced historical development and present situation, theory and solving method of the inverse scattering problem of acoustic wave, and the thesis study mainly the two dimensional inverse scattering problem of acoustic wave.First, for the direct scattering problem . in the paper, the double-single layer potential linear combination is take to be as the approximation solution of scattering wave of the Helmholtz equation for the Dirichlet boundary condition. The Nystrom method . Galerkin method and collation method are simply introduced respectively. These methods are used to solve the second kind of linear integral equation. As a result, the numerical solution of the direct scattering problem is obtained. Further, a comparison of these methods is also given by numerical examples.Second, using the result of direct scattering problem obtained above, a method to recover the impendence coefficient from the p-th coefficient of the far field pattern of the scattered wave for the impendence boundary condition is given. Since the problem is nonlinear and ill-posed. We turn it into a minimizing problem by using the Tikhonov regularization method. Then we use the qua.si-Newton method to solve the problem and finally give an approximation to the impendence coefficients, the convergence of this method is also proven. its numerical implementation show the efficiency of the inversion approach by comparing the recovered the impendence coefficients with the accurate one.Further, we introduced the Colt on and Monk method that was mentioned by Colton and Monk earlier for solving the inverse obstacle scattering problem, we compare it with Linear Sampling method that is used presently largely and point out the features of the two methods.Finally, using the results of solving the penetrable obstacle domain of direct scattering problem by the Tiklionov method.we give a linear sampling method to recover the penetrable boundary condition, by numerical examples, we also compare the recover domains with the accurate one and show the efficiency of the inversion approach.