The inverse backscattering problem and the inverse fixed angle scattering problem

该项目的主要目标是通过在介质边界上用机械产生的声波探测介质并测量边界上的介质响应，来研究介质(如地球)的声学性质的恢复。这个问题将从理论上进行研究，重点是从少量实验数据中恢复声学性质。这项研究的结果将在各种环境中提供洞察力，包括对其他行星次表面的探索以及医学成像环境，在这些环境中，可能无法在介质中的许多不同位置产生声波或在许多不同位置测量介质响应。该项目将专注于两个具体问题--恒速波动方程的潜在恢复后向散射问题和非恒定速度波动方程的定角速度恢复问题。对于具有位势的超双曲型方程，后向散射问题的量测与某些量测是等价的，因此我们将采用Bukhgeim-Klibanov方法来研究超双曲型方程相应的系数恢复问题。固定角度速度恢复问题将通过采用Bukhgeim-Klibanov方法来研究，该方法用于研究固定角度潜在恢复问题，并设计了新技术来克服与未知速度影响微分方程解结构相关的困难。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.