Formally determined inverse problems for hyperbolic PDEs

在声物理学中，人们感兴趣的是从声波探测时的介质响应恢复三维介质的声学特性。介质的性质由一个三变量函数来模拟，广义上称为势，声波满足波动方程，其中零阶项系数为势。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，在与入射波相同的方向上测量介质响应。测量来自所有可能方向的入射波的数据（后向散射数据），目标是恢复电位。PI建议研究特殊情况下，潜在的是分析的角度变量，通过适应沃尔泰拉型方法和使用尺度的Banach空间-一个想法使用罗曼诺夫在不同的反问题。PI还建议使用时域解释的后向散射问题，他已经成功地获得新的独特性结果，某些限制类的潜力。在第二个问题中，介质由放置在介质中心的点源激发，在足够长的时间内在边界处测量介质响应，并且目标再次是恢复电势。有一维版本的这个问题，即使与内部测量，这仍然是未解决的主要困难是，响应是在远离源点测量，使沃尔泰拉类型的方法，如层剥离或向下延续不适用。PI建议使用Ionescu和Klainerman的结果，在Carleman估计的帮助下研究这些问题，这些结果为构造径向Carleman权重提供了易于验证的条件。这项工作将对在地球物理学，石油勘探，医学成像，光纤以及任何希望从介质响应确定介质内部属性的领域工作的人感兴趣，在边界上测量的声波与在边界处产生的声波之间的关系。PI将监督研究生谁将对博士学位和可能的职业生涯在能源行业，医疗器械行业或学术界这些问题的工作。本科生可以在一个空间维度上研究这些问题的离散版本。这项工作的结果将通过研究生研讨会、在研究期刊上发表文章、在会议上介绍情况以及为非专业人员举办的介绍性讲习班来传播。

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.