Local Inverse Problems

首席研究员将研究地震成像，宇宙学和医学成像中出现的逆问题。第一个是从地震测量中恢复地球结构的数学问题。该项目考虑到地球最好被建模为弹性介质;它由几个核心组成，并且它是各向异性的（即，地震信号的速度可以取决于方向）。 第二个问题是宇宙学的应用：尽可能地从宇宙微波背景辐射测量中确定早期宇宙的阶段。最后，首席研究员将研究新的医学成像方法的数学，该方法使用两种不同的波来形成图像：一种波（例如电磁波）被发送到人体以激发细胞，这会产生另一种类型的波（例如声波），人们可以远离身体进行测量。在上述所有例子中，人们对使用局部信息解决局部问题特别感兴趣：当测量在局部完成时（比如，在相关域的边界的某个部分上），人们的愿望也是在局部恢复对象。事实上，在许多实际应用中，这种测量只能在局部进行，并且人们通常只对进行测量的点附近的区域中的被检查对象感兴趣。目标是了解测量数据中包含多少信息，确定这些信息对噪声和测量误差的敏感程度，并设计一种使用这些数据重建物体的方法。 更具体地说，该项目将通过以下途径进行研究：（1）从局部透镜/距离边界数据恢复边界接近严格凸边界点的紧致流形上的黎曼度规，直至等距;（2）从边界测量恢复洛伦兹度规，直至规范变换;（3）从边界上的波动方程数据恢复两种类型的度规;（4）求解具有分段光滑Lame参数的弹性物理模型的逆问题;（5）测地线和光线变换的逆;以及（6）求解医学成像中出现的逆源问题。这些问题包括线性和非线性问题，从适定性到轻度不适定性到不适定性。目的是恢复基本双曲方程中的主导项，这决定了几何形状。与该领域以前的许多工作不同，该项目允许共枕点的存在，尽管模一些额外的几何假设。在边界/透镜刚度的局部恢复黎曼度量的问题，直到等距，是非常具有挑战性的，因为简单的线性化不工作，并且存在固有的非唯一性。首席研究员希望结合联合收割机的想法，从张量层析成像，梅尔罗斯的散射演算，和其他新的想法来处理非线性。他的方法将证明全球唯一性和稳定性以及条件下，流形承认严格凸叶理。不排除共扼点的存在。时空中的反问题，包括积分几何问题，缺乏椭圆性，因为只有类空奇点是可恢复的。这使得问题不适定。光线变换是一种受限的光线变换，需要专门的微局部工具，甚至还没有完全开发用于非平坦度量。这些问题是感兴趣的几何（在那里他们被称为刚性问题），在部分理论的偏微分方程称为逆运动学和逆边值问题，并在应用程序，如地球物理。

The principal investigator will study inverse problems arising in seismic imaging, cosmology, and medical imaging. The first of these is the mathematical problem of recovering the structure of the Earth from seismic measurements. The project takes into account that the Earth is best modeled as an elastic medium; that it consist of several cores, and that it is anisotropic (i.e., that the speed of seismic signals can depend on the direction). A second problem is motivated by applications to cosmology: to determine the stage of the early Universe, to the extent possible, from the cosmic microwave background radiation measurements. Finally, the principal investigator will study the mathematics of new medical imaging methods that use two different waves to form an image: one wave (electromagnetic, for example) is sent to the human body to excite the cells, which creates another type of wave (acoustic, for example) that one measures away from the body. In all the aforementioned examples, one is especially interested in solving local problems using local information: when measurements are done locally (say, on some part of the boundary of the relevant domain), one's desire is to recover the object locally, as well. Indeed, in many practical applications such measurements can only be done locally, and very often one is interested in the object under scrutiny only in a region near the point where the measurement is taken. The goal is to understand how much information is included in the measured data, to determine how sensitive such information is to noise and measurement errors, and to devise a means of using this data to reconstruct the object. More specifically, the project will pursue the following avenues of research: (1) recovery of a Riemannian metric, up to isometry, on a compact manifold with boundary near a strictly convex boundary point from localized lens/distance boundary data; (2) recovery of a Lorentzian metric up to a gauge transformation from boundary measurements; (3) recovery of both types of metrics from wave equation data on the boundary; (4) solving the inverse problem for the elastic geophysics model with piecewise smooth Lame parameters; (5) inversion of the geodesic and the light ray transforms; and (6) solving inverse source problems that arise in medical imaging. Theses include both linear and nonlinear problems, from well-posed to mildly ill-posed to ill-posed. The aim is to recover the leading term in the underlying hyperbolic equation, which determines the geometry. Unlike much previous work in the area, this project allows for the existence of conjugate points, albeit modulo some additional geometric assumptions. The problem of local recovery a Riemannian metric in boundary/lens rigidity, up to isometry, is very challenging because simple linearization does not work, and there is inherent nonuniqueness. The principal investigator expects to combine ideas from tensor tomography, Melrose's scattering calculus, and other new ideas to deal with the nonlinearity. His methods would prove global uniqueness and stability as well, under the condition that the manifold admits a strictly convex foliation. Existence of conjugate points is not excluded. The inverse problems in time-space, including the integral geometry ones, lack ellipticity because only space-like singularities are recoverable. This makes the problem ill-posed. The light ray transform is a restricted ray transform, requiring specialized microlocal tools, not even fully developed yet for nonflat metrics. Those problems are of interest in geometry (where they are called rigidity problems), in the part of the theory of partial differential equations known as inverse kinematic and inverse boundary-value problems, and in applications like geophysics.