Hyperbolic Inverse Problems

在石油和天然气勘探、绘制行星内部地图或医学成像等领域，人们可以确定物体内部的性质，例如地球内部石油/天然气矿床的位置、确定行星内部组成的特征，或确定人体内部肿块是否癌变。由于在这些情况下钻孔或切割通常是昂贵的或不可行的，因此可以通过非侵入性方法探测这些物体，例如在物体边界上产生声波。期望的是，物体的内部组成将影响入射波和响应波，也只在物体的边界上测量，提供了一个进入物体内部的数学窗口。首席研究员（PI）将研究这种成像技术背后的数学原理。此外，PI将通过迷你课程、研讨会、个人对话和讲习班来培训这类数学的研究生和博士后。其中一些学生和博士后将利用这些技能为勘探石油、建造成像设备或涉及遥感的公司解决实际问题。像上面描述的问题，但与过度确定的数据，其中未知函数依赖的变量比数据少，已经得到了很多关注。PI关注较少研究的正式确定问题，其中未知函数依赖于与数据相同数量的变量。这类问题在两个或多个空间维度上比较困难，但在数据采集困难的情况下非常有用，对它们的研究是该领域的重要挑战之一。本项目将在三个空间维度上研究以下问题：定角散射问题、后向散射问题、点源问题和入射球面波问题。最近，PI和他的合作者使用了Bukhgeim-Klibanov方法的一种改进，证明了固定角散射问题的稳定性，并且证明了从点源问题和入射球面波问题中恢复给定系数的问题的唯一性。本项目旨在进一步调整Bukhgeim-Klibanov方法，使用Robbiano-Tataru型Carleman估计和唯一延拓论证来解决所提出的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In fields such as oil and gas prospecting, mapping the interior of a planet, or medical imaging, one determines properties of the interior of an object such as the location of oil/gas deposits in the interior of the earth, characterize the interior composition of a planet, or determine if an interior lump in the body is cancerous or not. Since drilling or cutting is often expensive or unfeasible in these situations, these objects are probed by non-invasive methods such as sound waves generated on the boundary of the object. The expectation is that the interior composition of the object will influence the incoming waves and the response wave, also measured only on the boundary of the object, provides a mathematical window into the interior of the object. The principal investigator (PI) will study the mathematics behind this imaging technique. Further, the PI will train graduate students and postdocs in this type of mathematics through mini-courses, seminars, personal conversations and workshops. Some of these students and postdocs will use these skills to solve practical problems for companies exploring for oil, building imaging devices, or involved in remote sensing.Problems like those described above, but with over-determined data, where the unknown function depends on fewer variables than the data, have received a lot of attention. The PI focuses on the less studied formally determined problems where the unknown function depends on the same number of variables as the data. Such problems, in two or more space dimensions, are harder but very useful in situations where data acquisition is difficult, and their investigation is one of the important challenges in the field. This project will study the following problems in three space dimensions: the fixed angle scattering problem, the backscattering problem, the point source problem, and the incoming spherical wave problem. Recently, using an adaptation of the Bukhgeim-Klibanov method, the PI and his collaborators proved stability for the fixed angle scattering problem for coefficients which are even in one of the variables and proved uniqueness for the problem of recovering a coefficient given data from the point source problem as well as the incoming spherical wave problem. This project aims at further adapting the Bukhgeim-Klibanov method to use Robbiano-Tataru type Carleman estimates and unique continuation arguments to tackle the proposed problems.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.