Electromagnetic Signatures of Inhomogeneities: Visibility vs. Invisibility

这个数学研究项目旨在提高对规则性，几何形状和不均匀性的材料含量之间的关系及其在电磁波探测时的可见性/不可见性的理解。研究人员计划开发工具来提高图像分辨率，并提供分析，以确定哪些几何特征更明显或更不明显。预计这项工作将对电磁成像技术产生直接影响，无论是涉及未知物体的雷达探测，材料成分的无损检测，还是活体器官的生物医学成像。同样的工具和分析也将用于电磁隐身。该项目将研究在多大程度上可以创建近似的斗篷，工作在宽频带的频率，同时使用材料，是物理现实。这将有助于更好地了解被动隐身装置的局限性及其物理实现性。研究者将指导博士后研究人员作为研究的一个组成部分。研究的目的是阐明一个不均匀体的规则性，几何形状和物质含量与其可见/不可见性之间的关系。这种理解可以用来更好地检测不均匀性或隐藏它们。该项目还旨在开发新的渐近公式，用于边界条件变化对小边界集的影响，这是优化设计的重要主题。该项目调查四个分专题：（1）规律性研究非散射波数的非均匀性（边界），（2）非均匀性几何的研究（3）最近关于完全隐身不可行性的结果与早期关于近似隐身可行性的结果的比较，（4）发展了“小”内部不均匀性和“小”边界（条件）不均匀性场效应的一致有效渐近公式。数学技术将包括尖锐的估计的影响，小的边界条件的极端对比度和速端曲线变换技术与相关的椭圆PDE估计应用于检查非散射不均匀性的规律性。为了开发极端对比度的边界条件不均匀性的小的相关措施，调查员将介绍和检查各种新颖的概念capacity.This award reflects NSF's法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This mathematics research project aims to improve understanding of the relation between the regularity, the geometry, and the material contents of an inhomogeneity and its visibility/invisibility properties when probed by electromagnetic waves. The investigator plans to develop tools to enhance image resolution and to provide analysis that determines which geometric features are more visible or less visible. It is anticipated that the work will have direct implications for electromagnetic imaging techniques, whether they concern radar detection of unknown objects, nondestructive testing of material components, or biomedical imaging of live organs. The same tools and analysis will also be brought to bear on electromagnetic cloaking. The project will examine to what extent one may create approximate cloaks that work at broad bands of frequencies and at the same time use materials that are physically realistic. This will provide a better understanding of the limitations of passive cloaking devices and their physical realizability. The investigator will mentor a postdoctoral researcher as an integral part of the research. The research is intended to elucidate the relation between the regularity, the geometry, and the material contents of an inhomogeneity and its visibility/invisibility properties. Such understanding can be used to better detect inhomogeneities or hide them. The project also aims to develop novel asymptotic formulas for the effect of a change in boundary conditions on small boundary sets, a topic of importance for optimal design. The project investigates four sub-topics: (1) study of regularity of (the boundary of) inhomogeneities that exhibit non-scattering wave numbers, (2) study of geometry of (the boundary of) inhomogeneities that exhibit in finitely many non-scattering wave numbers, (3) comparison of recent results about the infeasibility of perfect cloaking and earlier results about the feasibility of approximate cloaking, and (4) development of uniformly valid asymptotic formulas for the field effects of "small" internal inhomogeneities and "small" boundary (condition) inhomogeneities. The mathematical techniques will include sharp estimates of the effects of small boundary condition inhomogeneities of extreme contrast and hodograph transform techniques with associated elliptic PDE estimates applied to examine the regularity properties of non-scattering inhomogeneities. To develop the relevant measure of the smallness of boundary condition inhomogeneities of extreme contrast, the investigator will introduce and examine various novel notions of capacity.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.