Inverse problems for hyperbolic partial differential equations

当通过向物体发送波来探测物体以便从测量的响应中确定物体的某些性质时，就会出现一个典型的逆边值问题。建议的研究集中在使用声波方程对测量进行建模的情况。声学案例可以被视为几个地球物理成像问题的原型案例。例如，当目标是探测地质矿床时，这些问题可能是局部的，当目标是绘制地球内部结构图时，这些问题可能是全球性的。这项研究的目的是显著扩展反边值问题的数学理论，并弥合该理论与地球物理成像实际应用之间的差距。主要的理论目的是证明波动方程的反边值问题在完全坐标不变的情况下存在唯一解。主要的实用目标是发展一种新的反边值问题的计算方法。目前地球物理成像中使用的计算方法并不是基于边值反问题的数学理论。弥合理论和应用之间的差距有可能对后者产生很大影响，由此产生的物理理解也将指导进一步的数学研究。这项研究汇集了偏微分方程分析、微分几何和数值分析的思想。

A typical inverse boundary value problem arises when an object is probed by sending waves to it in order to determine some properties of the object from the measured responses. The proposed research focuses on the case where the measurements are modelled using the acoustic wave equation. The acoustic case can be seen as a prototype case for several geophysical imaging problems. These problems can be local, for example, when the goal is to detect geological deposits, or global when the goal is to chart the internal structure of the Earth. The aim of the proposed research is to significantly extend the mathematical theory of inverse boundary value problems and to bridge the gap between this theory and practical applications in geophysical imaging. The main theoretical objective is to show that the inverse boundary value problem for the wave equation has a unique solution in the fully coordinate invariant context. The main practical objective is to develop a new computational method for inverse boundary value problems. The presently used computational methods in geophysical imaging are not based on the mathematical theory of inverse boundary value problems. Bridging the gap between the theory and applications has a potential for a high impact on the latter, and the resulting physical understanding will also guide further mathematical study.The research brings together ideas from analysis of partial differential equations, differential geometry, and numerical analysis.