【作者】 袁敏；

【导师】 刘继军；

【作者基本信息】 东南大学 ， 应用数学， 2005， 硕士

【摘要】 逆散射问题是一类重要的反问题，其基本的任务是由散射波场的测量信息来探求散射体的物理性质，例如散射体的结构参数或者散射体的形状。数学上该类问题表示为椭圆型方程的系数(边界)反问题。探测法是逆散射问题中的一种重要的求解方法，它是由日本数学家M．Ikehata等人于1998年提出的。其主要思想是由散射波测量数据(如散射波的远场形式)构造一个带有散射体外面参数点的指示函数，当参数点靠近散射体的边界时，指示函数爆破，由此重建散射体的边界。其中核心的一步是由散射波的远场形式构造定义于包含散射体的一个已知区域Ω的边界上的Dirichlet-to-Neumann(D-to-N)映射，这是一个典型的不适定问题。本文的主要目的是对具有Sound-soft边界的散射体研究探测法的数值实现，主要的任务是考虑标志函数的构造，进而利用模拟数据检验探测法的实现效果，并探讨了可能存在的问题和解决的办法。为了更清楚地检验探测方法的数值结果，我们直接从Ω边界上的D-to-N映射来研究探测方法。我们的工作包含了如下三个方面。首先简单叙述了非连通区域上Helmholtz方程边值问题，即在散射体已知的假定下，在环形区域Ω＼(?)内用密度函数的方法求解正问题的解以得到反演输入数据。在探测法中重要的一步是Runge逼近函数的构造和数值实现，它在(?)Ω上的限制就是D-to-N映射，而这种定义于边界上的D-to-N映射就是需要的指示函数的输入数据。因此我们的第二个工作是讨论了Runge逼近函数的构造问题，它本质上是解第一类的积分算子方程，这也是不适定的。最后，我们讨论了所提出反演方法的数值实现，检验了探测方法的实现效果。更多还原

【Abstract】 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.更多还原