求解声波区域反散射问题的几种数值解法

【作者】 孟文辉；

【导师】 王连堂；

【作者基本信息】 西北大学 ， 应用数学， 2005， 硕士

【摘要】 反问题的研究是数学物理中一个较新的研究领域。声波反散射问题是一类典型的反问题,它在雷达和声纳等领域有广泛的应用前景。声波反散射问题被证明是不适定的,其求解起源于20世纪60年代中期Tikhonov的基础性论文。在这些论文中Tikhonov提出了求解不适定问题的方法—Tikhonov正则化方法。因此,这一方法被广泛的应用于声波反散射问题上。其中,声波散射问题是在入射波和散射区域等已知的条件下,求解其散射波和散射波对应的远场模式。反散射问题则是由散射方程的解来确定其散射区域或其它定解条件。本文对声波正散射问题和反散射问题都进行了研究,得到了很好的理论结果和数值结果。其中,反散射问题主要研究散射区域的反演。本文丰要作了以下工作: 1.对于较简单的积分方程反问题—线性矩问题,在Backus-Gilbert方法及修正方法的基础上,对修正方法进行了完善,并通过误差分析和数值例子表明了方法的易行性及精确性。 2.求解Dirichlet边界条件和阻尼边界条件下的声波正散射问题,对其解分别用双层位势和单层位势表示,并举出了数值例子进行求解,它是反问题研究的基础。 3.提出一种求解软散射区域的组合方法,它在波数k>0时是恒成立的,并给出了收敛性证明及数值例子。 4.用远场模式的完全及不完全数据对声波阻尼区域进行了反演,并进行了解的收敛性证明,举出了具有代表性的数值例子。 5.线性抽样(Linear Sampling)方法是求解声波反散射问题的最新方法,但是它的理论还很不完善。本文对阻尼区域反问题应用线性抽样方法进行求解,并对其解的收敛性及解的性质进行了理论上的证明,使得此方法得以完善,并通过数值例子表明了方法的易行性及定理的正确性。 6.提出近场数据的Linear Sampling方法,对其解的收敛性及其性质进行了分析,得出了与远场数据反演时相同的结论,并举出了数值例子。更多还原

【Abstract】 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.更多还原