Formally determined inverse problems for hyperbolic PDEs

在地球物理学中，人们感兴趣的是从声波探测到的介质响应中恢复三维介质的声学特性。介质的性质用三个变量的函数来描述，松散地称为势，声波满足以零阶项系数为势的波动方程。PI提出要研究两个具体问题。在第一个问题中，用平面波探测介质，并在足够长的时间间隔内沿与入射波相同的方向测量介质响应。这些数据是针对来自所有可能方向的入射波(后向散射数据)进行测量的，目标是恢复势能。PI建议通过采用Volterra型方法和使用Banach空间的尺度来研究势在角变量中解析时的特殊情况-Romanov在不同的反问题上使用的想法。PI还建议使用后向散射问题的时间域解释，他已经成功地使用该解释来获得某些受限制的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，介质的响应在边界上测量足够长的时间周期，目标同样是恢复势能。这个问题有一维版本，即使是内部测量，仍然没有解决，主要困难是在远离震源的点测量响应，因此Volterra类型的方法，如剥层或向下延拓不适用。PI建议利用Ionescu和Klanerman的结果借助Carleman估计来研究这些问题，这些估计给出了构造径向Carleman权重的容易验证的条件。这项工作将对从事地球物理、石油勘探、医学成像、光纤和任何希望通过在边界上测量的介质对在边界产生的声波的响应来确定介质内部性质的人感兴趣。PI将指导那些将致力于这些问题的研究生攻读博士学位，并可能在能源行业、医疗器械行业或学术界从事职业。本科生可以在一个空间维度上研究这些问题的离散版本。这项工作的成果将通过研究生研讨会、在研究期刊上发表文章以及在为非专家举办的会议和介绍性讲习班上的发言来传播。

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.