Local Inverse Problems

局部反问题

• 批准号: 1600327
• 负责人: Plamen Stefanov
• 金额: $ 37.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600327&HistoricalAwards=false
• 关键词: Local; Inverse; Problems

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.

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。第一个问题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是作为弹性介质建模；它由几个岩心组成，并且它是各向异性的（即地震信号的速度取决于方向）。第二个问题是由宇宙学的应用引起的：从宇宙微波背景辐射的测量中尽可能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原理，这种方法使用两种不同的波来形成图像：一种波（例如电磁波）被发送到人体来激发细胞，从而产生另一种波（例如声波波），人们可以在远离身体的地方测量到这种波。在上述所有示例中，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（例如，在相关领域边界的某些部分上），人们的愿望是局部恢复对象。事实上，在许多实际应用中，这种测量只能在局部进行，而且人们通常只对测量点附近的一个区域的被观察物体感兴趣。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对象的方法。更具体地说，该项目将追求以下研究途径：(1)从局部透镜/距离边界数据中恢复具有严格凸边界点附近边界的紧流形上的黎曼度量，直至等距；(2)从边界测量恢复到规范变换的洛伦兹度量；(3)从边界上的波动方程数据中恢复两类指标；(4)求解分段光滑Lame参数弹性地球物理模型的反演问题；(5)测地线反演与光线变换；(6)解决医学成像中出现的逆源问题。论文包括线性和非线性问题，从适定到轻度不适定再到不适定。目的是恢复基本的双曲方程的主导项，它决定了几何。与该领域以前的许多工作不同，该项目允许共轭点的存在，尽管模取了一些额外的几何假设。黎曼度量在边界/透镜刚度直至等距的局部恢复问题是一个非常具有挑战性的问题，因为简单的线性化不能工作，并且存在固有的非唯一性。首席研究员希望结合张量层析成像、梅尔罗斯散射演算和其他新思想来处理非线性。在流形允许严格凸叶化的条件下，他的方法也证明了全局唯一性和稳定性。不排除共轭点的存在性。时空反问题，包括积分几何反问题，都缺乏椭圆性，因为只有类空间奇点是可恢复的。这使得这个问题不适定。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。这些问题在几何中（它们被称为刚性问题），在偏微分方程理论的逆运动学和逆边值问题中，以及在地球物理学等应用中都很有趣。