Novel Sampling Methods for Electromagnetic Inverse Scattering Theory

逆电磁散射问题（IESP）的目的是根据物体散射的电磁场来确定未知物体。在过去的三十年里，IESP一直是工程、数学和物理界的一个活跃的研究课题，因为它对包括雷达、无损检测、医学成像和地球物理勘探在内的广泛应用产生了影响。然而，由于该问题通常是高度非线性和严重病态的，因此求解IESP是非常具有挑战性的。因此，尽管IESP的计算算法已经得到了广泛的研究，但对提高效率和鲁棒性的算法仍然有很高的要求。该项目通过为光学和雷达背景下的IESP开发新的高效采样型算法来满足这一需求。该项目还将涉及对计算数学的本科生和研究生的培训。解决IESP问题需要采用适当的采样方法来构造未知散射目标的近似指示函数。理想情况下，这些采样方法是快速的，非迭代的，并且不需要关于散射对象的先验信息。在本项目中，主要研究者和研究生将在不同类型的散射介质（包括无限周期介质、小和点状物体、有界非均匀介质和非线性介质）中为亥姆霍兹方程和麦克斯韦方程组开发新的采样型方法。除了上述特征之外，本项目将开发的采样型方法易于实现，并且对数据中的噪声具有极强的鲁棒性。本文将研究这些采样型方法的分辨率和稳定性分析，并通过实验数据对其进行验证。这些新方法有望为解决光学和雷达领域的IESP问题提供一种有前途的替代工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The inverse electromagnetic scattering problem (IESP) aims to determine an unknown object from the electromagnetic fields scattered by that object. The IESP has been an active research topic in the engineering, mathematics, and physics communities for the past three decades due to its impact on a wide range of applications including radar, nondestructive testing, medical imaging, and geophysical exploration. However, solving the IESP is very challenging since this problem is in general highly nonlinear and severely ill-posed. Therefore, although computational algorithms have been extensively studied for the IESP, there is still a high demand for algorithms with improved efficiency and robustness. The project addresses this demand by developing new and highly efficient sampling-type algorithms for the IESP in the context of optics and radar. This project will also involve the training of undergraduate and graduate students in computational mathematics. Solving the IESP involves proper sampling methods to construct an approximate indicator function for the unknown scattering object. Ideally these sampling methods are fast, non-iterative, and do not require a priori information about the scattering object. In this project, the principal investigator and graduate students will develop new sampling-type methods for the IESP for both the Helmholtz equation and the system of Maxwell's equations in different types of scattering media, including infinite periodic media, small and point-like objects, bounded inhomogeneous media, and nonlinear media. In addition to the aforementioned features, the sampling-type methods that will be developed in this project will be simple to implement and extremely robust against noise in the data. The resolution and stability analysis of these sampling-type methods and their validation by experimental data will be investigated. The new methods are expected to provide a promising alternative tool to solve the IESP in optics and radar.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.