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Several Numerical Solution Of Solving The Inverse Scattering Problem Of Acoustic Region

Posted on:2006-04-14Degree:MasterType:Thesis
Country:ChinaCandidate:W H MengFull Text:PDF
GTID:2190360155975734Subject:Applied Mathematics
Abstract/Summary:PDF Full Text Request
The field of inverse problem is a relatively new area of Mathematical research, having its origins in the the fundamental papers of Tikhonov in the mid-1960's. In these papers, Tikhonov introduced the regularization methods for linear ill-posed problems. It was realized that inverse problems were ill-posed. The inverse acoustic problem is a typical inverse problem. It has bright future in application on radar and sonar, etc. Before considering it we must have the results of direct problem. The direct problem is, given the "incident" waves and the scattering boundary, to find the "scattered" waves and in particular its behavior at large distances, i.e. its "far field pattern". The inverse acoustic problem takes this answer as its starting point and asks what is the incident waves and the scattering boundary, which give rise to such a far field behavior. In this paper, some problems of direct obstacle scattering and inverse scattering problems are investigated, some satisfactory conclusions are reached. The inverse scattering problem that we will mainly be concerned with in this paper is to determine the scattering boundary. Details are as follows:1. A modified method for linear moment problem, a kind of easier inverse problem, are improved. Numerical examples and error estimate are given showing accurate and simple of the method.2. As the bases of inverse problem, the direct scattering problem is solved by layer potential theory for Dirichlet and impedance boundary. The results of examples are given.3. Given a comprehensive method which suit for all wave number k (k > 0). The convergence of the method is proven. Numerical examples are given for Dirichlet boundary condition.4. A method is presented which use complete and incomplete data of the far field pattern of the scattering wave to recover the impedance boundary. The convergence of the method and numerical examples are also given.5. The linear sampling method is a new algorithm for solving the inverse scattering problem for acoustic waves. The method is based on showing that a linear integral equation of first kind-far field equation include a parameter z, which has a solution that becomes unbounded as z approaches the scattering boundary (?)D from inside D. Firstly, however, this equation is not unique solvable, then convergence of the solution is necessary. Secondly, as z ∈ R~2\(D|ˉ) the equation is unsolved, since the scattering D is unknown, this step is crucial for the linear sampling method. Base on the inverse impendence boundary problem these two problems were proven in this paper. Numerical examples show the theory is correct.6. Linear sampling method for near field data is presented. There are the same results as far field condition.
Keywords/Search Tags:Herglotz wave function, Layer potential, Linear moment, Inverse obstacle scattering, Far field pattern, Recover, Tikhonov regularization method, Far field equation, Linear sampling method, scattering boundary
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