Nonscattering Phenomena and Inverse Scattering

该项目侧重于研究逆散射和非散射现象，这些现象用于仅使用来自外部的散射波来确定物体或介质的特性，而无需物理探测物体或介质。逆散射在雷达和声纳成像、医学成像、地球物理勘探和天体物理学等领域有着广泛的应用。本项目旨在研究均匀自由空间中非均匀介质的非散射特性，并探索其在逆散射和隐身方面的潜在应用，从而使物体不被某些波检测或不受某些波的影响。此外，作为该项目的一部分，研究生和本科生将接受指导，这将增加他们的数学教育的深度和广度。尽管最近在一些特殊几何形状方面取得了进展，但一般非均匀性的非散射波数的存在与否在很大程度上仍然是一个悬而未决的问题。本项目旨在通过解决几个挑战来探索非均匀介质的非散射行为。这项工作的具体目标是：(1)表征非均匀性和波是非散射的必要条件，或者相反，是非平凡散射，(2)从数学上构建非散射介质和波，(3)建立一般非均匀性类的非散射波数的定量结果，(4)将非散射和/或非平凡散射的结果应用于逆问题，通过一次测量来唯一地确定未知介质的形状。该项目将阐明非散射现象，并为逆散射及相关领域的进展铺平道路。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。

This project focuses on the study of inverse scattering and non-scattering phenomena, which are used for determining the properties of an object or medium using only scattered waves from the exterior, without having to physically probe the object or medium. Inverse scattering has a wide range of applications in fields such as radar and sonar imaging, medical imaging, geophysical exploration, and astrophysics. This project aims to investigate non-scattering features of inhomogeneous media in a homogeneous free space and explore their potential applications to inverse scattering as well as invisibility, or cloaking, which can be used to make objects undetectable or unimpacted by certain waves. In addition, graduate and undergraduate student will be mentored as part of this project, which will add both depth and breadth to their mathematical education. Despite recent progress concerning some special geometries, the existence or nonexistence of nonscattering wavenumbers for general inhomogeneities largely remains an open question. This project aims to explore the non-scattering behavior of inhomogeneous media by addressing several challenges. The specific goals of this work are to (1) characterize necessary conditions for inhomogeneities and waves to be nonscattering, or the opposite, to be nontrivially scattering, (2) mathematically construct nonscattering media and waves, (3) establish quantitative results on nonscattering wavenumbers for a general class of inhomogeneities, and (4) apply results of nonscattering and/or nontrivial scattering to inverse problems for uniquely determining the shape of an unknown medium by one single measurement. This project will shed light on non-scattering phenomena and pave the way for advancements in inverse scattering and related fields.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.