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:
- 金额:$ 23万
- 依托单位:
- 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:
- 关键词:
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。
项目成果
期刊论文数量(5)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
The transmission problem in linear isotropic elasticity
线性各向同性弹性的传递问题
- DOI:10.2140/paa.2021.3.109
- 发表时间:2021
- 期刊:
- 影响因子:0
- 作者:Stefanov, Plamen;Uhlmann, Gunther;Vasy, András
- 通讯作者:Vasy, András
Sampling linear inverse problems with noise
对带有噪声的线性逆问题进行采样
- DOI:10.3233/asy-221795
- 发表时间:2023
- 期刊:
- 影响因子:1.4
- 作者:Stefanov, Plamen;Tindel, Samy
- 通讯作者:Tindel, Samy
The Radon transform with finitely many angles *
有限多个角度的 Radon 变换 *
- DOI:10.1088/1361-6420/acef53
- 发表时间:2023
- 期刊:
- 影响因子:2.1
- 作者:Stefanov, Plamen
- 通讯作者:Stefanov, Plamen
Inverse Boundary Problem for the Two Photon Absorption Transport Equation
二光子吸收输运方程的反边界问题
- DOI:10.1137/21m1417387
- 发表时间:2022
- 期刊:
- 影响因子:2
- 作者:Stefanov, Plamen;Zhong, Yimin
- 通讯作者:Zhong, Yimin
Recovery of a Cubic Non-linearity in the Wave Equation in the Weakly Non-linear Regime
弱非线性域中波动方程三次非线性的恢复
- DOI:10.1007/s00220-022-04359-0
- 发表时间:2022
- 期刊:
- 影响因子:2.4
- 作者:Sá Barreto, Antônio;Stefanov, Plamen
- 通讯作者:Stefanov, Plamen
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Plamen Stefanov其他文献
Quasimodes and resonances: Sharp lower bounds
准模和共振:尖锐的下界
- DOI:
10.1215/s0012-7094-99-09903-9 - 发表时间:
1999 - 期刊:
- 影响因子:2.5
- 作者:
Plamen Stefanov - 通讯作者:
Plamen Stefanov
Recovery of a general nonlinearity in the semilinear wave equation
半线性波动方程中一般非线性的恢复
- DOI:
- 发表时间:
2021 - 期刊:
- 影响因子:1.4
- 作者:
Antonio S'a Barreto;Plamen Stefanov - 通讯作者:
Plamen Stefanov
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction
恒定折射率的透射特征值的 Weyl 渐近
- DOI:
- 发表时间:
2013 - 期刊:
- 影响因子:0
- 作者:
Ha Pham;Plamen Stefanov - 通讯作者:
Plamen Stefanov
ON RANGE CONDITION OF THE TENSOR X-RAY TRANSFORM IN R
R 中张量 X 射线变换的在域条件
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:0
- 作者:
A. Denisiuk;Plamen Stefanov - 通讯作者:
Plamen Stefanov
Monitoring the surface states of a low-temperature carbon monoxide shift catalyst during operation
- DOI:
10.1016/s0166-9834(00)80431-5 - 发表时间:
1988-06-15 - 期刊:
- 影响因子:
- 作者:
Zˆarko Jovanović;Tsvetana Marinova;Plamen Stefanov - 通讯作者:
Plamen Stefanov
Plamen Stefanov的其他文献
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{{ truncateString('Plamen Stefanov', 18)}}的其他基金
Inverse Problems for Nonlinear Wave Phenomena
非线性波现象的反问题
- 批准号:
2154489 - 财政年份:2022
- 资助金额:
$ 23万 - 项目类别:
Standard Grant
US - Brazil Workshop on Scattering and Spectral Theory; Recife and Serrambi, Brazil
美国-巴西散射和光谱理论研讨会;
- 批准号:
0738079 - 财政年份:2008
- 资助金额:
$ 23万 - 项目类别:
Standard Grant
Collaborative Research: FRG: Inverse Problems in Transport Theory
合作研究:FRG:传输理论中的反问题
- 批准号:
0554065 - 财政年份:2006
- 资助金额:
$ 23万 - 项目类别:
Standard Grant
Inverse Anisotropic Problems and Resonances
逆各向异性问题和共振
- 批准号:
0400869 - 财政年份:2004
- 资助金额:
$ 23万 - 项目类别:
Standard Grant
相似海外基金
Inverse problems for degenerate hyperbolic partial differential equations on manifolds
流形上简并双曲偏微分方程的反问题
- 批准号:
22K20340 - 财政年份:2022
- 资助金额:
$ 23万 - 项目类别:
Grant-in-Aid for Research Activity Start-up
Geometric analysis of partial differential equations and inverse problems
偏微分方程和反问题的几何分析
- 批准号:
22K03381 - 财政年份:2022
- 资助金额:
$ 23万 - 项目类别:
Grant-in-Aid for Scientific Research (C)
Inverse Problems for Nonlinear Partial Differential Equations
非线性偏微分方程的反问题
- 批准号:
2111020 - 财政年份:2021
- 资助金额:
$ 23万 - 项目类别:
Standard Grant
Inverse problems for hyperbolic partial differential equations on Lorentzian manifolds
洛伦兹流形上双曲偏微分方程的反问题
- 批准号:
20J11497 - 财政年份:2020
- 资助金额:
$ 23万 - 项目类别:
Grant-in-Aid for JSPS Fellows
Stability analysis for inverse problems of fractional partial differential equations and related topics
分数阶偏微分方程反问题的稳定性分析及相关主题
- 批准号:
19K23400 - 财政年份:2019
- 资助金额:
$ 23万 - 项目类别:
Grant-in-Aid for Research Activity Start-up
Inverse problems with partial data
部分数据的反问题
- 批准号:
DP190103451 - 财政年份:2019
- 资助金额:
$ 23万 - 项目类别:
Discovery Projects
Partial Differential Equation Inverse Problems and Electrical Impedance Tomography
偏微分方程反问题和电阻抗断层扫描
- 批准号:
489653-2016 - 财政年份:2018
- 资助金额:
$ 23万 - 项目类别:
Alexander Graham Bell Canada Graduate Scholarships - Doctoral
Geometry of partial differential equations and inverse problems
偏微分方程的几何和反问题
- 批准号:
18H01126 - 财政年份:2018
- 资助金额:
$ 23万 - 项目类别:
Grant-in-Aid for Scientific Research (B)
Partial Differential Equation Inverse Problems and Electrical Impedance Tomography
偏微分方程反问题和电阻抗断层扫描
- 批准号:
489653-2016 - 财政年份:2017
- 资助金额:
$ 23万 - 项目类别:
Postgraduate Scholarships - Doctoral
Classification of Methods for Bayesian Inverse Problems Governed by Partial Differential Equations
偏微分方程治理贝叶斯反问题方法的分类
- 批准号:
1723211 - 财政年份:2017
- 资助金额:
$ 23万 - 项目类别:
Continuing Grant