Inverse Boundary Problems

该项目将解决在许多科学和技术领域出现的几个基本逆问题的数学理论，包括医学成像、地球物理学和无损检测。提出了四个主要的研究课题。首先是各向异性介质中的走时层析成像。在数学术语中，这涉及到从边界的测地线连接点的长度（行进时间）和其他运动学信息确定域内部的黎曼度量（各向异性声速）。第二个主题是电阻抗断层扫描（EIT），也被称为卡尔德隆问题。在这种相反的方法中，人们试图通过在介质的边界测量电压和电流来确定介质的导电性。第三个主题侧重于耦合物理或混合逆问题。在这类反问题中，人们试图通过物理原理结合两种波来确定介质的内部性质；也就是说，一种波具有高分辨率（如超声波），另一种波提供高对比度。例如光声层析成像（PAT），热声层析成像（TAT）和瞬态弹性成像（TE）。第四个主题是关于隐形和隐形：如何使物体对不同类型的波不可见。在反边界问题中，人们试图通过在介质边界处进行测量来确定介质的内部性质。换句话说，人们可以通过测量外部来“看到”内部是什么吗？CT扫描就是一个例子，这是一种常用的医学成像技术。一种是测量人体对x射线的反应，并根据这些信息绘制出人体内部的图像。该项目将研究新的医学成像技术，如光声断层扫描。这结合了一种成像方法的高分辨率和另一种成像方法的高对比度。一个重要的医学成像应用是乳腺癌检测。超声波提供高（亚毫米）分辨率，但对比度较低。另一方面，许多肿瘤从电磁波中吸收的能量比健康细胞多得多。光声断层扫描是将相对无害的光辐射送入组织。这会引起加热（随着温度在毫开尔文范围内的升高），从而产生传播超声波（光声效应）。该项目将探索的另一个领域是通过测量地震波穿越地球所需的时间来绘制地球内部的图像。通过这种方式，人们尝试在地震提供的信息的帮助下穿越地球中心。该项目的最后一个主要研究课题是隐形和隐形的研究。一个人能使物体对光、声和其他类型的波不可见吗？

This project will address the mathematical theory of several fundamental inverse problems arising in many areas of science and technology, including medical imaging, geophysics, and nondestructive testing. Four major topics of research are proposed. The first is travel-time tomography in anisotropic media. In mathematical terms this involves the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joining points of the boundary (travel times) and from other kinematic information. The second topic is electric impedance tomography (EIT), also called Calderon's problem. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at its boundary. The third topic focuses on coupled-physics or hybrid inverse problems. In this type of inverse problem one attempts to determine the internal properties of a medium by combining two types of waves through a physical principle; namely, one wave that has high resolution (e.g., ultrasound) and another that provides high contrast. Examples are photoacoustic tomography (PAT), thermoacoustic tomography (TAT), and transient elastography (TE). The fourth topic is on invisibility and cloaking: how to make objects invisible to different types of waves.In inverse boundary problems one attempts to determine the internal properties of a medium by making measurements at the boundary of the medium. In other words, can one "see" what is inside by making measurements on the outside? An example is a CT scan, a commonly used medical imaging technique. One measures the response of the body to X-rays and makes an image of what is inside from this information. The project will investigate new proposed medical imaging techniques, such as photoacoustic tomography. This combines the high resolution of one imaging method with the high contrast capabilities of another. One important medical imaging application is breast cancer detection. Ultrasound provides a high (sub-millimeter) resolution, but suffers from low contrast. On the other hand, many tumors absorb much more energy from electromagnetic waves than do healthy cells. Photoacoustic tomography consists of sending relatively harmless optical radiation into tissues. This causes heating (with increases of the temperature in the millikelvin range), which results in the generation of propagating ultrasound waves (the photo-acoustic effect). Another area that the project will explore is to image the Earth's interior by measuring the time that it takes for seismic waves to traverse it. In this way one attempts a journey through the center of the Earth with the help of information provided by earthquakes. A final major topic of research in the project is the study of cloaking and invisibility. Can one make objects invisible to light, sound, and other types of waves?