Inverse Boundary Problems

在反边界问题中，人们试图通过在介质的边界上进行测量来确定介质的内部性质。换句话说，一个人能通过测量外面的东西来“看到”里面是什么吗？CT扫描就是一个例子，这是一种常用的医学成像技术。其中一个人测量身体对X射线的反应，并根据这些信息制作内部信息的图像。首席研究人员将研究使用高频声波代替X射线的超声波传输和反射层析成像。此外，还将考虑宇宙学中的几个逆问题。主要的问题是，人们能否通过在地球附近进行的测量来确定数十亿年前的宇宙结构。首席研究员将讨论一些基本逆问题的数学理论，这些问题出现在许多科学和技术领域，包括医学成像、地球物理、宇宙学和无损检测。将调查三个主要研究课题。第一种是各向异性介质中的旅行时层析成像。在数学术语中，这涉及到根据连接边界点的测地线的长度(旅行时间)和其他运动学信息来确定区域内部的黎曼度量或芬斯勒度量(各向异性声速)。第二种是在广义相对论、弹性力学、流体力学等许多应用中产生的非线性方程的反问题。其主要思想是利用波的非线性相互作用产生新的波，这将有助于解决反问题。第三种是电阻抗断层成像(EIT)，也被称为卡尔德隆问题。在这种逆方法中，人们试图通过在边界上进行电压和电流测量来确定介质的导电性。在该项目中，准线性案例将被考虑以及分数拉普拉斯的类似问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In inverse boundary problems one attempts to determine the internal properties of a medium by making measurements at the boundary of the medium. In other words, can one "see" what is inside by making measurements on the outside? An example are CT scans, a commonly used medical imaging technique. One measures the response of the body to X-rays and makes an image of what is inside from this information. The principal investigator will investigate ultrasound transmission and reflection tomography which uses high frequency sound waves instead of X-rays. Also several inverse problems in cosmology will be considered. The main questions is whether one can determine the structure of the Universe billions of years ago from measurements made near the Earth.The principal investigator will address the mathematical theory of several fundamental inverse problems arising in many areas of science and technology including medical imaging, geophysics, cosmology, and nondestructive testing. Three major topics of research will be investigated. The first one is Travel Time Tomography in anisotropic media. In mathematical terms this involves the determination of a Riemannian metric or Finsler metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joining points of the boundary (travel times) and from other kinematic information. The second is inverse problems for non-linear equations arising in many applications including general relativity, elasticity, fluids etc. The main idea is to use the nonlinear interaction of waves to produce new waves that will help solve the inverse problems. The third is Electric Impedance Tomography (EIT), also called Calderon's problem. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at the boundary. In the project the quasilinear case will be considered as well as an analog problem for the fractional Laplacian.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.