无界界面分层介质中电磁波逆散射问题的理论与数值方法研究

Inverse electromagnetic scattering problems are one of the most rapidly developing areas of applied mathematics in the last three decades, are also a very active research field of the inverse problems for the partial differential equations in recent years, and often occur in many branches of science and engineering such as nondestructive testing, medical imaging, radar and sonar, geological exploration, remote sensing. The project intends to study the following problems: first, we will establish the correspondence between the near-field data and parts of the far-field data, and establish a mixed reciprocity relation between the near field from point source and the far field from the plane waves, on the basis of the knowledge of the two relations, we will study the uniqueness for the inverse electromagnetic scattering problems in a stratified medium with unbounded interface by using the singularity of point source; then, by using the ways of the quantitative methods such as linear sampling method, singular sources method and factorization method, and by using the far-field data, we will propose some numerical inversion methods to efficiently reconstruct the location, the shape and the physical properties of the scatterer, and determine the refractive index. We hope that this project can enrich the theoretical and numerical methods for the inverse electromagnetic scattering problems, and provide some new research ways for solving inverse problems.

电磁波逆散射问题是近三十年来应用数学中发展最迅速的领域之一，也是近年来偏微分方程反问题领域中一个非常活跃的研究热点，在无损探伤、医学诊断、雷达与声纳、地质勘探、遥感技术等方面有广泛的应用。本项目拟研究以下问题：1、建立近场数据和部分远场数据之间的一一对应关系，构造点源散射得到的近场和平面波散射得到的远场之间的混合交互关系，基于这两个关系，利用点源的奇异性，证明无界界面分层介质中电磁波逆散射问题的唯一性理论；2、借鉴线性采样法、奇异源法和分解法等各种定量方法的思路，设计快速有效的数值方法，利用远场作为测量数据来反演散射体的位置、形状及其物理性质，以及确定折射率，并利用若干数值例子加以验证。通过本课题的研究，建立若干有创新性的研究电磁波逆散射问题的理论和数值方法，为解决相关的反问题提供新的研究思路和研究途径。

电磁波逆散射问题是近三十年来应用数学中发展最迅速的领域之一，也是近年来偏微分方程反问题领域中一个非常活跃的研究热点，在无损探伤、医学诊断、雷达与声纳、地质勘探和遥感技术等方面有广泛的应用。本项目研究了以下问题：1、在三维柱面对称波导的电磁波逆散射问题中，借鉴线性采样法的思路，利用远场作为测量数据建立一个关于Green函数的积分方程，此方程正则解的范数作为指示函数来反演未知散射体的位置及形状；2、在二维水平无限长波导的电磁波逆散射问题中，证明了逆散射问题的唯一性理论，同样借鉴线性采样法的思路，利用近场作为测量数据建立一个关于点源的积分方程，此方程正则解的范数作为指示函数来反演未知散射体的位置、形状及其物理性质，并利用若干数值例子加以验证。通过本课题的研究，进一步丰富了研究电磁波逆散射问题的理论和数值方法，为实际应用提供了数学理论依据和可靠的数据参考。