介质不连续性的反演方法及其数值实现

项目名称： 介质不连续性的反演方法及其数值实现

项目编号： No.11301075

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 王海兵

作者单位： 东南大学

项目金额： 22万元

中文摘要： 利用外部测量数据探测介质的内部结构是一类重要的数学物理反问题，当介质内部具有某种不连续性时，这类问题数学上变得特别困难。波场逆散射和热传导是重构介质不连续性(如异物、空腔和裂缝等)的两种被广泛采用的检测模型，数学上分别对应于椭圆型和抛物型偏微分方程反问题。本项目研究两个方面的内容：其一，以阻尼型柱状散射体的斜入射电磁波逆散射为模型，在探测方法的框架下，利用斜导数边值问题Green函数的奇异性和逐点估计，研究由单一的电场或磁场测量数据重构散射体几何形状及边界阻尼系数的正则化方法及其数值实现，并分析入射角度及阻尼系数对边界重建精度的影响；其二，以介质热传导为模型，研究利用温度场的边界测量数据重建介质内部未知异物几何形状的线性抽样方法，核心是运用抽象Cauchy问题的可解性理论，建立抛物方程內透射问题解的存在唯一性。本项目的研究将为其它基于偏微分方程的介质成像问题提供新的数学方法。

中文关键词： 反问题；逆散射；热成像；数值方法；

英文摘要： Detecting interior structure of the medium from exterior measurements is an important inverse problem of mathematical physics.When the medium has discontinuities, this problem becomes quite difficult. Inverse scattering of wave fields and heat conduction are two widely applied models of reconstructing the medium discontinuities, such as cavities, inclusions or cracks. Mathematically, they are formulated as inverse problems for partial differential equations of elliptic and parabolic types. This project is concerned with the following two problems. First, we consider an inverse scattering problem for obliquely incident electromagnetic waves by an impedance cylinder. In the framework of the probe method, we construct the Green function of the oblique boundary value problem governing the direct scattering problem, and analyze its singularity with point-wise estimate. Then we develop a regularization method for reconstruction of the scatterer and the boundary impedance from measurement data of electric or magnetic field. Moreover, we will also analyze the influences of the incident angle and the impedance coefficient on the accuracy of the numerical reconstructions. Second, based on the model of heat conduction in medium, we study the linear sampling-type method for reconstructing unknown inclusions inside a heat c

英文关键词： Inverse problems；Inverse scattering；Thermal imaging；numerical methods；