Inverse Problems: Visibility and Invisibility

理解我们周围世界的一些最强大的方法使用波传播的形式：电磁波(例如光或无线电)；声波；弹性波等等，因为它们可以以一种可以理解的方式与不同的介质相互作用，而对这些介质施加的影响很小。这些传感方法的非侵入性应用于许多科学技术领域，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反问题的主要方面，而主体的另一个方面试图通过波来隐藏物体，即使其“看不见”。这些项目涉及到各种波浪的可见性和不可见性两个方面。为了提高可见性，研究人员计划开发数学技术，以解决反问题中的一些具有挑战性的问题，这些问题在未来可能会有现实世界的应用。例如，热声断层扫描(TAT)医学成像设备有可能在比目前可行的更早的阶段检测到乳腺癌细胞。由这种耦合物理模式导致的对比度和分辨率的同时提高是深入数学研究的主题。另一个例子是CT扫描，它的数学理论是积分几何的基本工具。分析波在复杂材料中的传播数学对于解释有关地球的地震信息是必不可少的。至于隐身，“超材料”的发展允许电磁或声学介质的定制化，并启发了基于变换光学的隐形斗篷设计理论。在数学层面上，这导致了基于电磁波(例如，光)操纵的进一步应用。这些项目将专注于几个问题，即人造黑洞和伊顿透镜。该项目的第一个主要主题是麦克斯韦方程的反边值问题。需要研究的具有挑战性的问题包括：具有不完全测量数据的电磁反问题和具有各向异性参数的介质上的电磁反问题。我们还将考虑描述一类弯曲加速波包的麦克斯韦方程的一些特解。该项目的这一部分还涉及电磁不可见性和其他基于变换光学的应用，如人造黑洞。PI还提议研究另一种基于奇异几何的隐形装置的构造想法，即众所周知的伊顿透镜。该项目的第二个主要主题集中在非线性波传播中出现的反问题，其中非线性提供了新的信息。这也自然而然地引出了项目的第三个主题，因为双曲波传播和几何边界刚性问题之间的联系。这些反问题构成了几种层析成像方法的基础。PI计划在更普遍的情况下和有限的测量下就这一主题开展工作。在该项目的最后部分，PI计划考虑耦合物理(即混合)医学成像模式。特别是，PI对从内部吸收辐射图重建电属性的唯一性感兴趣。

Some of the most powerful methods of understanding the world around us use forms of wave propagation: electromagnetic waves (e.g. light or radio); sound waves; elastic waves and so on, because they can interact with different media in an understandable way while exerting little influence on those media. The non-intrusive nature of these sensing methods is applied in many areas of science and technology including medical imaging, geophysics, non-destructive testing, remote sensing and so on. The fundamental mathematical theory behind such enhancement of "visibility" is the main aspect of inverse problems, while another aspect of the subject seeks to hide an object from detection by waves, that is, to make it "invisible". These projects concern both aspects, visibility and invisibility, for various waves. For visibility, the investigator plans to develop mathematical techniques to address some of the challenging questions in inverse problems that may 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 is currently feasible. The simultaneous improvement in contrast and resolution resulting from such coupled-physics modalities is the subject of intensive mathematical study. Another example is the CT scan, whose mathematical theory is the fundamental tool in integral geometry. Analysis of the mathematics of wave propagation in complicated materials is essential for interpreting seismic information about the earth. As for invisibility, the development of "meta-materials" permits the customization of electromagnetic or acoustic media and has inspired the theory of transformation-optics based invisible cloak design. At the mathematical level, this leads to further applications based on electromagnetic waves (e.g., light) manipulation. These projects will focus on a couple of issues, known as the artificial blackhole and Eaton lens. The first major topic of the project is inverse boundary value problems for Maxwell's equations. Challenging questions to be investigated include: inverse electromagnetic problems with incomplete data of measurements and inverse electromagnetic problems on a medium with anisotropic parameters. We will also consider some special solutions to Maxwell's equations that describe a type of bending accelerating wave packets. This part of the project also concerns electromagnetic invisibility and other applications based on transformation-optics, such as artificial black holes. The PI is also proposing to investigate another construction idea of cloaking device based on singular geometry, known as the Eaton lens. The second major topic of the project focuses on inverse problems arising in nonlinear wave propagation, where the nonlinearity provides new information. This also leads naturally to the third topic of the project due to the connection between hyperbolic wave propagations and geometric boundary rigidity problems. These inverse problems form the basis of several tomography methods. The PI plans to work on this topic for more generalized cases, and with limited measurements. In the last part of the project the PI plans to consider coupled-physics (i.e., hybrid) medical imaging modalities. In particular, the PI is interested in the uniqueness of the reconstruction of electrical properties from the internal absorbed radiation map.