Electromagnetic Signatures of Inhomogeneities: Visibility vs. Invisibility

这项数学研究项目旨在提高对电磁波探测时不均匀的规律性、几何和物质含量与其可见性/不可见性之间的关系的理解。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，哪些更不明显。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形。该项目将检查一个人可以在多大程度上创造出在宽频带工作的近似斗篷，同时使用物理上逼真的材料。这将使人们更好地了解被动隐形装置的局限性及其物理实现能力。作为研究的组成部分，研究人员将指导一名博士后研究员。这项研究旨在阐明非均质性的规律性、几何和材料含量与其可见性/不可见性之间的关系。这样的理解可以用来更好地检测不均质性或隐藏它们。该项目还旨在为边界条件变化对小边界集的影响开发新的渐近公式，这是优化设计的一个重要主题。该项目研究了四个子主题：(1)表现出非散射波数的不均匀(边界)的规律性研究，(2)有限多个非散射波数表现出的不均匀(边界)的几何研究，(3)关于完美伪装不可行性的最新结果与早期关于近似伪装可行性的结果的比较，以及(4)发展“小的”内部不均匀和“小的”边界(条件)不均匀的场效应的一致有效的渐近公式。数学技术将包括对小边界条件的影响的精确估计，极端对比度的不均匀性，以及速度图变换技术和相关的椭圆PDE估计，用于检查非散射不均匀的正则性。为了发展极端对比的边界条件不均匀小的相关衡量标准，调查员将引入和检查各种新的能力概念。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.