Inverse Problems for Nonlinear Wave Phenomena

非线性波现象的反问题

基本信息

批准号：
2154489
负责人：
Plamen Stefanov
金额：
$ 43.23万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154489&HistoricalAwards=false
关键词：
Inverse Problems Nonlinear Wave Phenomena

项目摘要

The project will study nonlinear wave phenomena and related inverse problems. Of particular interest are topics related to nonlinear optics, including the static (DC) and the optical (AC) Kerr effects, where the medium changes its index of refraction either under the influence of a strong external electric field or by self-modulation. The project will study nonlinear acoustics and nonlinear elasticity as well. One of the main goals is to understand the underlying models and the solutions well in the asymptotic high-frequency regime. The next goal is to solve the associated inverse problems: to determine the parameters of the medium from remote measurements. Ultrasound and elastography are known to work in the nonlinear regime, which is one of the motivations. In addition, the project will study the propagation of singular waves in case of a linear elastic-fluid interaction across an interface, and the associated inverse problem of recovery the parameters of the parameters of the medium, inspired by geophysical applications. The project will also study a question on the proper discretization of the geodesic X-ray transform and similar Radon transforms. The project will provide research training opportunities for graduate students.The investigator plans to analyze propagation of waves for the nonlinear wave fundamental phenomena: nonlinear optics, acoustics and elasticity. Such phenomena are described by quasilinear partial differential equations, for which the geometric optics theory is less than complete. The PI plans to exploit the specific nature of those models, combined with physics intuition. The PI plans to explain the DC and the AC Kerr effects for the nonlinear Maxwell equations, in particular, and to find the right asymptotic regimes under which they occur: the relationship between the wavelength and the intensity of the waves. Then the PI will study the associated inverse problems of recovery of the parameters of the medium from remote observations. The PI also plans to further study propagation of elastic and pressure waves in liquid-solid media: reflection, transmission and mode conversion across a smooth surface separating the two media. This is inspired by the model of Earth, where the Crust and the Mantle are solid, the Upper Core is believed to be liquid but the core is still solid. Finally, the PI plans to study sampling and proper discretization and inversion of the geodesic X-ray transform and similar Radon type of transforms. This is related to the recent work by the PI connecting sampling theory with semiclassical analysis, relating the sampling requirements to the semi-classical wave front set rather than looking into the “band limit” (the support of the Fourier transform).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.

该项目将研究非线性波现象和相关的反演问题，特别感兴趣的是与非线性光学相关的主题，包括静态（DC）和光学（AC）克尔效应，其中介质在影响下改变其折射率。该项目还将研究非线性声学和非线性弹性，下一个目标是很好地理解渐近高频区域的基础模型和解决方案。解决相关的反问题：通过远程测量确定介质参数已知超声波和弹性成像在非线性状态下工作，这是该项目的动机之一。此外，该项目还将研究奇异波的传播。受地球物理应用的启发，该项目还将研究测地线 X 射线变换的适当离散化问题和跨界面的线性弹性流体相互作用，以及恢复介质参数的相关逆问题。相似的该项目将为研究生提供研究培训机会。研究人员计划分析非线性波基本现象的传播：非线性光学、声学和弹性，这些现象用拟线性偏微分方程来描述，其中几何。光学理论并不完整。 PI 计划结合物理直觉来解释这些模型的具体性质，特别是解释非线性麦克斯韦方程组的 DC 和 AC Kerr 效应。然后，PI 将研究从远程观测中恢复介质参数的相关逆问题。液固介质中的弹性波和压力波：在分隔两种介质的光滑表面上的反射、传输和模式转换这受到地球模型的启发，其中地壳和地幔是固体，而上核被认为是固体。液体但核心仍然存在最后，PI 计划研究测地线 X 射线变换和类似 Radon 类型的变换的采样以及适当的离散化和反演，这与 PI 将采样理论与半经典分析联系起来的最新工作有关。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。