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，因为它们可以以可理解的方式与不同的介质相互作用，而对这些介质几乎没有影响。这些非侵入性的传感方法应用于许多科学和技术领域，包括医学成像、生物物理学、非破坏性测试、遥感等。这种增强“可见性”背后的基本数学理论是反问题的主要方面，而该学科的另一个方面是寻求隐藏物体，使其不被波探测到，即使其“不可见”。这些项目涉及两个方面，可见性和不可见性，为各种波。对于可见性，研究人员计划开发数学技术，以解决一些具有挑战性的问题，在反问题，可能有真实的世界的应用在未来。例如，热声断层扫描（达特）医学成像模式有可能在比目前可行的更早的阶段检测乳腺癌细胞。同时改善对比度和分辨率所产生的这种耦合物理方式是密集的数学研究的主题。另一个例子是CT扫描，其数学理论是积分几何的基本工具。波在复杂材料中传播的数学分析对于解释地球的地震信息是必不可少的。至于隐形，“超材料”的发展允许电磁或声学介质的定制，并激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的进一步应用（例如，光）操纵。这些项目将集中在一对夫妇的问题，被称为人造黑洞和伊顿透镜。该项目的第一个主要课题是麦克斯韦方程组的逆边值问题。研究的问题包括：不完全测量数据的电磁场反问题和各向异性介质上的电磁场反问题。我们还将考虑描述一种弯曲加速波包的麦克斯韦方程组的一些特殊解。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形状的隐形装置的构造思想，称为伊顿透镜。该项目的第二个主要主题集中在非线性波传播中出现的逆问题，其中非线性提供了新的信息。由于双曲波传播和几何边界刚性问题之间的联系，这也自然导致了该项目的第三个主题。这些逆问题形成了几种层析成像方法的基础。PI计划在更广泛的情况下研究该主题，并且测量有限。在项目的最后一部分，PI计划考虑耦合物理（即，混合）医学成像模态。特别是，PI感兴趣的是从内部吸收辐射图的电气特性的重建的唯一性。

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.