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.

在边界反问题中，人们试图通过在介质的边界处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部来“看到”内部吗？ 一个例子是CT扫描，一种常用的医学成像技术。一种是测量身体对X射线的反应，并根据这些信息制作出身体内部的图像。主要研究者将研究使用高频声波而不是X射线的超声透射和反射断层扫描。 也将考虑宇宙学中的几个逆问题。主要的问题是，人们是否可以确定宇宙的结构，数十亿年前从地球附近的测量。首席研究员将解决几个基本的逆问题的数学理论，在许多科学和技术领域，包括医学成像，电子物理学，宇宙学和无损检测。将研究三个主要课题。第一个是各向异性介质中的走时层析成像。在数学术语中，这涉及到从连接边界点的测地线的长度（旅行时间）和其他运动学信息确定域内部的黎曼度量或芬斯勒度量（各向异性声速）。第二个是在许多应用中出现的非线性方程的反问题，包括广义相对论，弹性，流体等的主要思想是使用波的非线性相互作用，以产生新的波，这将有助于解决反问题。第三个是电阻抗断层成像（EIT），也称为Calderon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，准线性的情况下，将被视为以及模拟问题的分数拉普拉斯。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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

On an inverse boundary value problem for a nonlinear elastic wave equation

非线性弹性波方程的反边值问题

• DOI: 10.1016/j.matpur.2021.07.005
• 发表时间: 2021
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Uhlmann, Gunther;Zhai, Jian
• 通讯作者: Zhai, Jian