权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Inverse Problems in Partial Differential Equations and Geometry

偏微分方程和几何中的反问题

基本信息

批准号：
1900475
负责人：
Plamen Stefanov
金额：
$ 23万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900475&HistoricalAwards=false
关键词：
Inverse Problems Partial Differential Equations

项目摘要

The primary focus of this project is on inverse problems of determining parameters of media from remote measurements, in particular those arising in Earth exploration, in cosmology and imaging of moving and changing media, in medical imaging, and the sampling theory of linear and non-linear inverse problems. The principal investigator will study the elastic earth model with coefficients jumping across smooth surfaces and propagation and mode conversion of pressure and shear waves in it. The goal is to show that one can recover those coefficients stably from travel times of seismic waves. The principal investigator will study problems of recovery of moving media and Lorentzian metrics arising in relativity from remote observations. Tomography problems arising in medical imaging will be studied as well. Finally, the principal investigator will study the problem of finding the optimal sampling rate of linear and non-linear operators with applications to Inverse Problems. The interest to those is motivated by the fact that real life measurements are discrete, they typically average over small detectors, and numerical simulations are done on discrete grids as well. The principal investigator plans to do a full analysis of propagation, reflection, transmission and mode conversion of elastic pressure and shear waves (singularities) in isotropic elasticity with variable coefficients jumping across smooth surfaces modeling the boundaries between the Mantle, etc. Some of this analysis has been done in the flat constant coefficient case. The principal investigator will study Rayleigh and Stoneley surface waves as well. Next, the principal investigator will analyze the tensor tomography problem in Lorentzian geometry and the non-linear problem of recovery of a Lorentzian metric from remote observations. This problem is harder than its Riemannian counterpart because signals moving faster than light are unrecoverable in a stable way. The tomography problems arising in medical imaging, including Compton camera imaging, will be treated as Fourier Integral Operators and its analysis would require specific tools from that calculus. Finally, the principal investigator plans to study sampling of images of linear and non-linear operators motivated by practical considerations of ability to measure discrete data only. The principal investigator will connect sampling theory with semiclassical analysis and instead of looking into the "band limit" (the support of the Fourier transform) he will relate the sampling requirements to the semi-classical wave front set.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.

该项目的主要重点是从遥测确定介质参数的反问题，特别是在地球探测、宇宙学和运动和变化介质的成像、医学成像以及线性和非线性反问题的采样理论中出现的反问题。主要的研究人员将研究具有跨越光滑表面的系数的弹性地球模型以及压力波和横波在其中的传播和模式转换。其目的是证明人们可以从地震波的旅行时稳定地恢复这些系数。首席研究员将研究运动介质的恢复问题和从远程观测中产生的相对论洛伦兹度量。还将研究医学成像中出现的层析成像问题。最后，主要研究人员将研究寻找线性和非线性算子的最优采样率的问题，并将其应用于反问题。人们之所以对此感兴趣，是因为现实生活中的测量是离散的，它们通常是通过小型探测器进行平均的，数值模拟也是在离散的网格上进行的。主要的研究人员计划对各向同性弹性中的弹性压力波和横波(奇点)的传播、反射、传输和模式转换进行全面的分析，其中一些分析是在平坦的常系数情况下进行的。首席研究员还将研究瑞利和斯通利表面波。接下来，主要研究人员将分析洛伦兹几何中的张量层析成像问题，以及从远程观测中恢复洛伦兹度量的非线性问题。这个问题比黎曼的问题更困难，因为移动速度超过光速的信号无法以稳定的方式恢复。医学成像中出现的层析成像问题，包括康普顿相机成像，将被视为傅立叶积分算子，其分析将需要来自该微积分的特定工具。最后，首席研究员计划研究线性和非线性算子的图像采样，其动机是实际考虑到仅测量离散数据的能力。首席研究员将把抽样理论与半经典分析联系起来，而不是研究“波段极限”(傅立叶变换的支持)，他将把抽样要求与半经典波前集联系起来。这一奖项反映了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。