Inverse Boundary Problems

提出者将解决在反问题研究中出现的基本研究问题。科学的根本任务是探索我们周围的世界。要做到这一点，最有效的方法是向物体发送波并测量其响应。反问题在于从这些观测中确定被探测对象的性质。这些问题出现在科学和应用的所有领域，在这些领域中，需要确定预期或观察到的结果的原因。这项拟议的研究考虑了反问题数学理论中的基本问题，这些问题的动机是医学成像、地球物理和遥感等领域的应用。解决这些难题的数学技术的发展无疑将在未来造福人类的现实世界中得到应用。例如，X射线层析成像的数学对CT扫描和其他医学成像技术的发展做出了贡献，这些技术有助于拯救日常生命。建议的研究还将包括研究生作为课程工作和博士论文的一部分，以及博士后研究员。具体地说，建议者计划四个主要研究主题。第一种是电阻抗层析成像。在这种逆方法中，人们试图通过在边界上进行电压和电流测量来确定介质的导电性。这一逆方法已被认为是一种有价值的医学诊断工具，用于乳腺癌的早期检测。第二个是关于如何将物体“隐形”的问题，即如何使其在电磁波和其他类型的波中不可见，这是近年来受到极大关注的一个课题。第三种是旅行时间层析成像，它通过测量地震产生的地震波的旅行时间来确定地球的内部结构。最后一个主题是逆散射，它涉及远场测量，例如在遥感、声纳和雷达中出现。

The proposer will address basic research questions arising in the study of Inverse Problems. The fundamental task of science is to probe the world around us. The most powerful method to do this is to send waves to an object and measure the response. Inverse Problems consist in determining properties of the object being probed from these observations. These problems arise in all fields of science and applications where causes for a desired or observed effect are to be determined. The proposed research considers fundamental questions in the mathematical theory of inverse problems which are motivated by applications to medical imaging, geophysics, and remote sensing among others. Development of mathematical techniques to address these difficult questions will undoubtedly have real world applications in the future that will benefit mankind. The mathematics of X-ray tomography has, for instance, contributed to the development of CT scans and other medical imaging techniques that help save lives everyday. The proposed research will also involve graduate students as part of their course work and PhD theses, and also postdoctoral fellows.Specifically, the proposer plans four major topics of research. The first one is Electric Impedance Tomography. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at the boundary. This inverse method has been proposed as a valuable diagnostic tool in medicine for the early detection of breast cancer. The second is on the subject on how to "cloak" objects, that is how to make them invisible to electromagnetic waves and other type of waves, a subject that has received a lot of attention in recent years. The third is Travel Time Tomography that arises in determining the inner structure of the Earth by measuring the travel times of seismic waves produced by earthquakes. The final topic is inverse scattering that involves far field measurements and arises, for instance, in remote sensing, sonar and radar.