Inverse problems with partial data

这个项目的目的是使用数学，特别是微观局部分析的理论，来确定一个人为了通过医学成像中常用的一些断层成像方法重建图像所需的测量量。该项目的预期结果包括显示任意小的一组边界测量足以重建各种重要的偏微分方程的系数，例如薛定谔方程、狄拉克算子和麦克斯韦方程。除了提供一个可以建立数值算法的理论基础外，这个项目还将提供反问题和独特的连续理论之间缺失的环节。这项研究的下游影响将导致更高效和更准确的层析成像方法，可以在一系列成像应用中实施。

This project aims to use mathematics, in particular the theory of micro-local analysis, to determine the amount of measurements one needs in order to reconstruct an image by some of the tomography methods commonly used in medical imaging. Expected outcomes of this project include showing that an arbitrarily small set of boundary measurements is sufficient to reconstruct the coefficients of various important partial differential equations such as Schrodinger equation, Dirac operators, and Maxwell equations. In addition to providing a theoretical foundation upon which one can build numerical algorithms, this project will also provide the missing link between inverse problems and unique continuation theory. The downstream impact of this research will lead to more efficient and accurate tomography methods which can be implemented in a range of imaging applications.