The inverse backscattering problem and the inverse fixed angle scattering problem

逆后向散射问题和逆固定角散射问题

• 批准号: 2307800
• 负责人: Rakesh Rakesh
• 金额: $ 13.24万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307800&HistoricalAwards=false
• 关键词: inverse; backscattering; problem; fixed; angle; inverse; backscattering; problem; fixed; angle

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.

该项目的主要目的是通过探测介质在介质边界上机械生成的声波并测量边界上的介质响应的声波来研究介质（例如地球）的声学特性的回收。理论上将研究这个问题，重点是从少数实验的数据中恢复声学特性。这项研究的这一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设置，在媒介中可能无法在许多不同位置在许多不同位置产生声波或在许多不同位置的培养基响应。速度波方程。反向散射问题测量值等于具有潜力的超纤维方程的某些测量值，因此，将通过适应Bukhgeim-Klibanov方法来研究超纤维方程的相应系数恢复问题。固定角度恢复问题将通过适应Bukhgeim-Klibanov的适应来研究用于研究固定角度潜在恢复问题的方法，并设计了新技术来克服与未知速度相关的困难，影响了差异方程的解决方案的结构。这些奖项反映了NSF的构建范围和构建的依据，该奖项通过构建的构建范围依赖于宽广的依据。 标准。