Inverse Boundary Problems

PI: Gunther Uhlmann， University WashingtonDMS-0245414-------------------------------------------------------------------------摘要提出者将解决医学成像、地球物理和光学成像中出现的几个基本逆问题。模拟当前物理情况的方程组是电导率方程、相空间中的一类哈密顿方程、弹性动力学系统和麦克斯韦方程组。反问题是通过在边界处进行测量，分别确定物体或物体内部的电导率、哈密顿系统、弹性参数和电磁参数。我们提出四个主要的研究课题。第一个是电阻抗断层扫描。在这种相反的方法中，人们试图通过在边界处测量电压和电流来确定介质的导电性。第二个主题是从边界连接点的测地线长度（行进时间）确定域内部的黎曼度量（各向异性声速）。第三个主题是地震成像。我们计划使用微局部分析方法来生成波动方程解，这将有助于对地球上地壳的非均质性和各向异性进行成像。最后一个主题是电磁系统和麦克斯韦方程组。特别是，我们建议利用电磁场可以极化的事实来确定电磁参数，如手性，它测量材料分子组成的不对称性。反问题出现在科学和应用的所有领域，在这些领域中，需要确定期望的或观察到的效果的原因。通过解决反声学问题，我们了解了地球的内部结构；通过解决反x射线衍射问题，我们了解了DNA的结构；通过研究物质被粒子轰击时的散射，我们了解了原子及其组成部分的结构。还有许多其他具有实际重要性的反问题的例子。材料科学提供了一类例子。一项重要的任务是在不破坏材料的情况下找出材料内部的成分。医学诊断提供了另一个逆向问题至关重要的领域。在日常生活中作为诊断工具使用的x射线断层扫描和CAT扫描是特别熟悉的逆方法的例子。在x射线断层扫描中，高频x射线穿过人体，然后根据记录的信息确定其密度。拉东在本世纪初开发了计算密度所需的数学方法。关于反边界问题的建议的目标是为地球物理学、医学成像和光学成像中出现的几个反问题开发数学重建公式，类似于Radon为x射线断层成像开发的公式。

PI: Gunther Uhlmann, University of WashingtonDMS-0245414------------------------------------------------------------------------- ABSTRACTThe proposer will address several fundamental inverse problems arising in medical imaging, geophysics, and optical imaging. The systems of equations modeling the physical situation at hand are the conductivity equation, a class of Hamiltonian equations in phase space, the system of elastodynamics and Maxwell's equations. The inverse problems consist in determining the interior conductivity, the Hamiltonian system, the elastic and electromagnetic parameters, respectively, in the interior of a body or object by making measurements at the boundary. We propose 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. The second topic is the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joiningpoints of the boundary (travel times). The third topic is seismic imaging. We plan to use the methods of microlocal analysis to generate wave-equation solutions that will help to image the heterogeneity and anisotropy of the Earth's upper crust. The last topic is ElectromagneticSystems and Maxwell's equations. In particular we propose to utilize the fact that electromagnetic fields can be polarized to determine electromagnetic parameters like chirality, which measures the asymmetry in the molecular makeup of a material.Inverse problems arise in all fields of sciences and applications where causes for a desired or observed effect are to be determined. We know about the interior structure of the Earth by solving inverse acoustic problems, the structure of DNA from solving inverse X-ray diffractionproblems, and the structure of the atom and its constituents from studying the scattering when materials are bombarded with particles. There are numerous other examples of inverse problems of practical importance. Material science provides one class of examples. One important task is tofigure out what is inside a material without destroying it in the process. Medical diagnosis provides another area where inverse problems are of vital importance. X-ray tomography and CAT scans, which are used in everyday life as diagnostic tools, are particularly familiar examples of inverse methods. In X-ray tomography high frequency X-rays are sent through the body and then the density is determined from the recorded information. The mathematics needed to recover the density was developed by Radon in the early part of the century. The goal of the proposal on inverse boundary problems is to develop mathematical reconstruction formulas for several inverse problems arising in geophysics, medical imaging and optical imaging, analog to the one developed by Radon for the case of X-ray tomography.