Inverse Problems: Visibility and Invisibility

点击翻译按钮获取中文摘要

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.