复杂介质中障碍反散射问题的唯一性及数值算法

Inverse electromagnetic scattering problems are a clss of important inverse problems for partial differential equations, which has wide applications in the fields such as seismic exploration, material science and radar. The main difficulties in solving this kind of inverse problems is due to the fact that these problem are not only nonlinear but also ill-posed. The project studies the inverse scattering problems by bounded obstacles buried in a complex medium background, which aims to obtain the uniqueness results and numerical algorithms from the knowledge of far-field or near-field data. In order to address these issue, the essential difficulty due to the unknown inhomogeneous medium background needs to be overcome. The project focuses on the studies on the mathmatical theory and methods motivated by practical applications, consisting of the model, analysis and computation, which is of critical importance, not only to the research on mathematics, but also to the implementation of numeical algorithms

电磁反散射问题是一类重要的偏微分方程反问题，其难点在于问题的非线性和不适定性，在地质勘探、材料科学、雷达等领域有广泛的应用。本项目拟对两类具有明确应用背景的复杂介质中的障碍反散射问题进行研究，目标是建立基于远场或近场数据反演的唯一性理论和数值算法，重点解决未知且非均匀的背景介质带来的本质困难。本项目是应用数学驱动的数学理论和方法的研究，集建模、分析与计算于一体，无论是对数学研究本身还是对算法的数值实现，都具有重要的意义。

本项目主要围绕非均匀介质中障碍散射问题及其反问题进行了理论与算法研究. 对于时域散射问题，提出了构造三维时域电磁散射问题PML的新方法并证明了指数收敛性，相关工作已经发表在SIAM J. Numer. Anal.，这是三维电磁散射问题PML方法方面的第一个收敛性结果. 对于时谐反散射问题，提出了证明反问题唯一性的新方法，基于远场或近场测量证明了含有掩埋物体的界面反演的唯一性及流固耦合反问题的唯一性，并进一步基于Dirichlet-to-Neumann映射证明了双参数反演的局部唯一性结果，上述结果已发表在Inverse Problems, Inverse Problems and Imaging及Journal of Differential Equations等期刊上. 对于数值算法方面，提出了重建复杂曲面的几个有效的非迭代型采样方法，相关工作已接受或发表在SIAM J. Sci. Comput., SIAM J. Imaging. Sci.及Inverse Problems and Imaging上. 上述研究成果有望为雷达成像、医学成像、无损探测等领域提供重要的理论支撑与依据。