The inverse backscattering problem and the inverse fixed angle scattering problem

本项目的主要目的是通过在介质的边界上机械产生的声波探测介质并测量边界上的介质响应来研究恢复介质（例如地球）的声学特性。这个问题将在理论上进行研究，重点是从来自少量实验的数据中恢复声学特性。这项研究的结果将在各种环境中提供洞察力，包括探索其他行星的地下以及医学成像环境，在这些地方，可能无法在介质的多个不同位置产生声波，或在多个不同位置测量介质的响应。该项目将集中于两个具体问题：常速度波动方程的势恢复后向散射问题和非常速度波动方程的固定角速度恢复问题。后向散射问题的测量结果等价于超双曲型方程的某些测量结果，因此超双曲型方程的相应系数恢复问题将通过Bukhgeim-Klibanov方法的一个改进来研究。固定角速度恢复问题将通过用于研究固定角势恢复问题的Bukhgeim-Klibanov方法的修改来研究，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识产权进行评估来支持。优点和更广泛的影响审查标准。

The main goal of this project is to study the recovery of acoustic properties of a medium, such as the earth, by probing the medium with sound waves mechanically generated on the boundary of the medium and measuring the medium response on the boundary. This question will be studied theoretically with a focus on the recovery of the acoustic property from the data coming from a small number of experiments. This results of this study will provide insight in a variety of settings, including the exploration of the subsurface of other planets as well as medical imaging settings, where it may not be possible to generate acoustic waves at many different locations in the medium or measure the medium response at many different locations.The project will focus on two specific problems - the potential recovery backscattering problem for the constant velocity wave equation and the fixed angle velocity recovery problem for the non-constant velocity wave equation. The backscattering problem measurements are equivalent to certain measurements for the ultra-hyperbolic equation with potential, so the corresponding coefficient recovery problem for the ultra-hyperbolic equation will be studied by an adaptation of the Bukhgeim-Klibanov method. The fixed angle velocity recovery problem will be studied by an adaptation of the Bukhgeim-Klibanov method used to study the fixed angle potential recovery problem, with new techniques devised to overcome the difficulties associated with the unknown velocity affecting the structure of the solution of the differential equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.