Inverse Boundary Problems

逆边界问题

• 批准号: 1800453
• 负责人: Gunther Uhlmann
• 金额: $ 27万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800453&HistoricalAwards=false
• 关键词: Inverse; Boundary; Problems

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.

在逆边界问题中，人们试图通过在介质的边界处进行测量来确定培养基的内部特性。换句话说，一个人可以通过在外部进行测量来“看到”内部的内容吗？ 一个例子是CT扫描，一种常用的医学成像技术。一个人可以衡量身体对X射线的响应，并从此信息中构成内部内容的图像。主要研究者将研究使用高频声波而不是X射线的超声传输和反射断层扫描。 还将考虑宇宙学中的几个反问题。主要问题是，是否可以通过在地球附近进行的测量来确定数十亿美元的宇宙结构。 ，宇宙学和无损测试。将研究三个主要研究主题。第一个是各向异性媒体中的旅行时间断层扫描。用数学术语，这涉及确定域内的Riemannian公制或Finsler度量（各向异性音速），从地球学的长度上连接了边界（行程时间）和其他运动信息。第二个是在许多应用中引起的非线性方程的反问题，包括一般相对论，弹性，流体等。主要思想是使用波浪的非线性相互作用来产生新的波浪，以帮助解决逆问题。第三个是电阻纸层析成像（EIT），也称为卡尔德隆的问题。在这种逆方法中，人们试图通过在边界处进行电压和电流测量来确定介质的电导率。在该项目中，准线性案件将被视为分数拉普拉斯的一个模拟问题。该奖项反映了NSF的法定任务，并且使用基金会的智力优点和更广泛的影响审查标准，认为值得通过评估来获得支持。

项目成果

Reconstruction of the Collision Kernel in the Nonlinear Boltzmann Equation

非线性玻尔兹曼方程碰撞核的重构

• DOI: 10.1137/20m1329366
• 发表时间: 2021
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Lai, Ru-Yu;Uhlmann, Gunther;Yang, Yang
• 通讯作者: Yang, Yang

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

共形横向各向异性流形上 Calderón 问题的重构

• DOI: 10.1016/j.jfa.2021.109191
• 发表时间: 2021
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Feizmohammadi, Ali;Krupchyk, Katya;Oksanen, Lauri;Uhlmann, Gunther
• 通讯作者: Uhlmann, Gunther

Convolutional Neural Networks in Phase Space and Inverse Problems

相空间和逆问题中的卷积神经网络

• DOI: 10.1137/19m1294484
• 发表时间: 2020
• 期刊: SIAM Journal on Applied Mathematics
• 影响因子: 1.9
• 作者: Uhlmann, Gunther;Wang, Yiran
• 通讯作者: Wang, Yiran

Travel Time Tomography in Stationary Spacetimes

静止时空中的走时断层扫描

• DOI: 10.1007/s12220-021-00620-5
• 发表时间: 2021
• 期刊: The Journal of Geometric Analysis
• 作者: Uhlmann, Gunther;Yang, Yang;Zhou, Hanming
• 通讯作者: Zhou, Hanming

Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

• DOI: 10.4310/mrl.2020.v27.n6.a10
• 发表时间: 2019-09
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Katya Krupchyk;G. Uhlmann