Hyperbolic Inverse Problems

在诸如石油和天然气勘探、绘制行星内部或医学成像等领域中，人们确定物体内部的性质，诸如地球内部的石油/天然气沉积物的位置，表征行星的内部成分，或确定体内的内部肿块是否是癌性的。由于在这些情况下钻孔或切割通常是昂贵的或不可行的，因此通过非侵入性方法（例如在物体的边界上产生的声波）来探测这些物体。期望的是，物体的内部组成将影响入射波，并且也仅在物体的边界上测量的响应波提供了进入物体内部的数学窗口。主要研究者（PI）将研究这种成像技术背后的数学。此外，PI将通过微型课程，研讨会，个人对话和研讨会来培训研究生和博士后。这些学生和博士后中的一些人将使用这些技能来解决公司勘探石油，建造成像设备或参与遥感的实际问题。像上面描述的那些问题，但具有超定数据，未知函数依赖于比数据更少的变量，已经受到了很多关注。PI侧重于研究较少的正式确定的问题，其中未知函数依赖于与数据相同数量的变量。这些问题，在两个或更多的空间维度，是困难的，但非常有用的情况下，数据采集是困难的，他们的调查是在该领域的重要挑战之一。本计画将在三维空间中研究下列问题：固定角散射问题、后向散射问题、点源问题及入射球面波问题。最近，使用Bukhgeim-Klibanov方法的适应，PI和他的合作者证明了固定角度散射问题的稳定性，即使是在一个变量中的系数，并证明了从点源问题以及传入球面波问题中恢复给定数据的系数问题的唯一性。该项目旨在进一步调整Bukhgeim-Klibanov方法，以使用Robbiano-Tataru型Carleman估计和独特的连续参数来解决拟议的问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.