障碍与介质混合反散射问题的数学理论和数值算法

The two basic problems in classical scattering theory are the scattering of time-harmonic waves by a bounded impenetrable obstacle and by a penetrable inhomogeneous medium with compact support. In the past thirty years, there has been an extensive study in these two problems and rich results can be found in the literature. However, the unknown object may also be a mixed type scatterer which is given as the union of a bounded impenetrable obstacle and a penetrable inhomogeneous medium with compact support. The research of this model plays an extremely important role in such different areas in radar, remote sensing, nondestructive testing and medical imaging. ..This project is concered with the mathematical theory and numerical methods for solving the inverse scattering by mixed scatterers. The main context of this project includes the following two mathematical questions: (1) Uniqueness and stability. Uniqueness means whether the location, shape and physical properties of the mixed scatterer can be determined by the measurement date, while the stability analysis explains the ill-posedness of the inverse problems. (2) Algorithms for both direct and inverse scattering by mixed scatterers. We mainly study the sampling methods and iterative methods for location and shape reconstruction. Efficient and stable algorithms will also be developed to distinguish between an impenetrable obstacle and a penetrable inhomogeneous medium. Finally, Some numerical examples are presented to demonstrate the feasibility and effectiveness of our methods.

经典的时谐波反散射问题中未知散射体有两类：有界不可穿透障碍和具有紧支集的可穿透非均匀介质。近三十年，数学家对这两类基本反散射问题进行了深入研究并取得了丰富的成果。然而，未知散射体也可能是不可穿透障碍与非均匀介质组成的混合散射体。这类混合散射体反散射的研究在雷达探测、遥感、无损探伤和医学成像等科学工程领域中具有极其重要的应用价值。.本项目将考虑混合散射体反散射问题的数学理论和数值算法。主要研究内容包括：（1）唯一性和稳定性理论。唯一性探索测量数据是否可以唯一确定混合散射体的位置、形状及其物理性质，而稳定性分析则揭示这类反问题的不适定程度。 （2）混合散射体散射问题及其反问题的算法设计和数值实施。为重构混合散射体的位置和形状，我们主要考虑采样法和迭代法。我们也将设计快速有效区分不可穿透障碍和可穿透非均匀介质两大分支的数值算法，并通过若干数值例子加以验证。

反散射问题是近三十年来应用数学与计算数学中一个非常活跃的研究领域。在实际反散射应用中，除了散射体位置形状的信息未知外，散射体的物理性质往往也是不确定的。特别是散射体可能有很多分支，而各个分支的物理性质也可能是不同的。本项目旨在研究混合反散射问题的理论和数值方法。. (1)反散射问题的唯一性理论. 我们讨论了非均匀介质中的散射问题,并证明即使背景介质未知,散射体也可以被唯一确定,所用的数据为一个入射方向的平面波对应的多频远场模式.这个结果也被推广到电磁场情形..（2）混合反散射问题的稳定算法及数值实现.我们设计了两类直接采样算法, 一类基于单频的多方向远场数据, 一类基于多频的单方向远场数据.两种算法都具有快速,稳定和有效的特点.. (3)基于不完备数据的唯一性和算法.我们主要讨论了缺失数据的完备化及其在反散射问题中的应用.特别,这类算法不依赖散射体的先验信息.