Formally determined inverse problems for hyperbolic PDEs

双曲偏微分方程的正式确定的反问题

• 批准号: 1312708
• 负责人: Rakesh Rakesh
• 金额: $ 8.83万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312708&HistoricalAwards=false
• 关键词: Formally; determined; inverse; problems; hyperbolic

In geophysics one is interested in recovering the acoustic properties of a three dimensional medium from the medium response when probed by an acoustic wave. The property of the medium is modeled by a function of three variables, loosely called the potential, and the acoustic wave satisfies the wave equation with the zeroth order term coefficient being the potential. The PI proposes studying two specific problems. In the first problem, the medium is probed by a plane wave and the medium response is measured in the same direction as the incoming wave, over a long enough time interval. This data is measured for incoming waves from all possible directions (the back-scattering data) and the goal is to recover the potential. The PI proposes studying the special case when the potential is analytic in the angular variables, by adapting Volterra type methods and using scales of Banach spaces - an idea used by Romanov on a different inverse problem. The PI proposes also using a time domain interpretation of the back-scattering problem which he has already used successfully to obtain new uniqueness results for certain restricted classes of potentials. In the second problem, the medium is excited by a point source placed at the center of the medium, the medium response is measured at the boundary for a long enough time period, and the goal, again, is the recovery of the potential. There are one dimensional versions of this problem, even with interior measurements, which remain unsolved with the chief difficulty being that the response is measured at points away from the source so that Volterra type methods such as layer stripping or downward continuation are not applicable. The PI proposes studying these problems with the help of Carleman estimates using results of Ionescu and Klainerman which give easily verifiable conditions for constructing radial Carleman weights.This work will be of interest to people working in geophysics, oil prospecting, medical imaging, fiber-optics, and in any area where one wishes to determine the properties of the interior of a medium from the response of the medium, measured on the boundary, to an acoustic wave generated at the boundary. The PI will supervise graduate students who will work on these problems towards a PhD and possible careers in the energy industry, the medical devices industry or in academia. Undergraduate students may study discrete versions of these problems in one space dimension. The results of this work will be disseminated through graduate seminars, publications in research journals, and presentations at conferences and introductory workshops for non-specialists.

在地球物理学中，人们感兴趣的是从声波探测时的介质响应中恢复三维介质的声学特性。介质的性质是用三个变量的函数来表示的，粗略地称为势，声波满足以零阶项系数为势的波动方程。PI建议研究两个具体问题。在第一个问题中，用平面波探测介质，在足够长的时间间隔内测量与入射波方向相同的介质响应。该数据测量来自所有可能方向的入射波（后向散射数据），目标是恢复势能。PI建议通过采用Volterra型方法和使用Banach空间的尺度（Romanov在另一个反问题上使用的思想）来研究角变量中势是解析的特殊情况。PI还建议使用后向散射问题的时域解释，他已经成功地使用该解释获得了某些受限势类的新的唯一性结果。在第二个问题中，介质被放置在介质中心的点源激发，在边界处测量足够长的时间内的介质响应，目标，同样，是恢复势。这个问题有一维的版本，即使是内部测量，仍然没有解决，主要的困难是响应是在远离源的点上测量的，因此Volterra类型的方法，如层剥离或向下延拓不适用。PI建议利用Ionescu和Klainerman的结果，利用Carleman估计来研究这些问题，这些结果给出了构造径向Carleman权值的容易验证的条件。这项工作将对从事地球物理学、石油勘探、医学成像、光纤和任何希望根据在边界上测量的介质对边界上产生的声波的响应来确定介质内部性质的领域的人们感兴趣。PI将指导研究生，这些学生将在这些问题上取得博士学位，并可能在能源行业、医疗设备行业或学术界从事职业。本科生可以在一个空间维度上学习这些问题的离散版本。这项工作的成果将通过研究生讨论会、研究期刊出版物、会议报告和非专业人员介绍性讲习班来传播。