Electromagnetic Inverse Problems: Visibility and Invisibility

本项目旨在解决由亥姆霍兹方程和麦克斯韦方程模拟的电磁波传播反问题数学理论中的几个基本问题。该项目的第一部分涉及通过在边界处进行电磁测量来确定介质内部性质的问题。具有挑战性的问题包括从边界数据中恢复电磁包裹体和障碍物的形状信息，在边界数据不完整的各向同性介质中完全重建电磁参数，以及各向异性介质的逆问题，其中只有在变量变化的情况下才能进行唯一的重建。在后一种情况下，这种独特性的丧失导致了该项目的第二个主要主题，即基于变换光学的不可见性。提出的研究涵盖了正则化近似电磁隐身方案，其极限行为为理想隐身设计所需的奇异结构提供了重要的物理和数学意义。拟议项目的最后一部分考虑热声断层扫描（TAT），这是一种混合医学成像方式，通过物理“光声”效应将低频电磁波与声波结合在一起。特别是，主要研究者计划解决从内部吸收辐射图重建介质的电学参数和折射率的问题。科学的基本任务是探索我们周围的世界。要做到这一点，最有力的工具是使用波（例如电磁波、声波、弹性波），因为波可以以非侵入性的方式与不同的介质相互作用。这种非侵入性的调查在许多科学和技术领域都非常受欢迎，包括医学成像、地球物理、无损检测、遥感等。这种“可见性”增强背后的基本数学理论是逆问题的主要主题，而另一个方面是隐藏物体不被波探测，即使其“不可见”。该项目涉及电磁波的可见性和不可见性两个方面。为了可见性，提议者计划开发数学技术来解决逆问题中的一些具有挑战性的问题，这些问题无疑将在未来的现实世界中得到应用。例如，热声断层扫描（TAT）医学成像模式有可能在比目前可行的更早的阶段检测到乳腺癌细胞。TAT数学理论的发展也将揭示更大类别的内部数据逆问题，如PAT（光声断层扫描），TE（瞬态弹性成像），UMT（超声调制电和光阻抗断层扫描），以及其他类似的混合成像模式。同时提高对比度和分辨率是密集的数学研究的主题，也是这个项目的主题。至于隐形，具有自然界中不存在的电磁或声学特性的“超材料”的发展激发了基于变换光学的隐形斗篷蓝图。实验和理论结果表明，隐形不再只是科幻小说。本理论研究试图在一个现实的近似方案下理解和面对困难，这将有利于实验的努力。它提供了一种有效的方式来弥合现实与想象之间的差距，这对于当前的技术来说仍然是巨大的。

This project aims to address several fundamental problems in the mathematical theory of inverse problems involving electromagnetic wave propagation as modeled by the Helmholtz and Maxwell equations. The first part of the project concerns the question of determining the internal properties of a medium by making electromagnetic (EM) measurements at the boundary. Challenging questions include recovering the shape information of electromagnetic inclusions and obstacles from boundary data, full reconstruction of EM parameters for an isotropic medium with incomplete boundary data, and the inverse problem for anisotropic media, where the unique reconstruction is possible only up to a change of variables. Such loss of uniqueness in the latter case leads to the second major topic of the project, namely, transformation-optics-based invisibility. The proposed study covers the regularized approximate electromagnetic cloaking scheme, whose limiting behavior provides important physical and mathematical implications for the singular structure required in an ideal cloaking design. The last part of the proposed project considers thermo-acoustic tomography (TAT), a hybrid medical imaging modality that combines low frequency electromagnetic waves with acoustic waves through the physical "photo-acoustic' effect. In particular, the principal investigator plans to address the problem of reconstructing the electrical parameters and refractive indices of the medium from the internal absorbed radiation map. The fundamental task of science is to probe the world around us. The most powerful tool to do this is to use waves (e.g., electromagnetic waves, acoustic waves, elastic waves), because waves can interact with different media in nonintrusive ways. Such nonintrusive investigation is greatly appreciated in many areas of science and technology, including medical imaging, geophysics, nondestructive testing, remote sensing, and so on. The fundamental mathematical theory behind such enhancement of "visibility" is the main topic of inverse problems, while another aspect is to hide an object from detection by waves, that is, to make it "invisible." The proposed project concerns both aspects, visibility and invisibility, for electromagnetic waves. For visibility, the proposer plans to develop mathematical techniques to address some of the challenging questions in inverse problems that will undoubtedly have real world applications in the future. For instance, the thermo-acoustic tomography (TAT) medical imaging modality has the potential to detect breast cancer cells at a much earlier stage than what is currently feasible. The development of mathematical theory of TAT will also shed light on a larger class of inverse problems with internal data such as PAT (photo-acoustic tomography), TE (transient elastography), UMT (ultrasound modulated electrical and optical impedance tomography), as well as other similar hybrid imaging modalities. The simultaneous improvement in contrast and resolution is the subject of intensive mathematical study and also of this project. As for invisibility, the development of "meta-materials" that have electromagnetic or acoustic properties not found in nature has inspired the transformation-optics based blueprint of the invisible cloak. Experimental and theoretical results indicate that invisibility is no longer just science fiction. This theoretical study seeks to understand and confront the difficulties under a realistic approximate scheme, which will benefit experimental endeavors. It provides an effective way to bridge the gap between reality and imagination that is still vast for current technology.