Inverse Problems in Partial Differential Equations and Geometry

该项目的主要重点是从远程测量中确定介质参数的反问题，特别是在地球勘探、宇宙学和运动和变化介质成像、医学成像以及线性和非线性反问题的抽样理论中产生的反问题。主要研究光滑表面上系数跳跃的弹性地球模型及其中压力波和剪切波的传播和模态转换。我们的目标是证明人们可以从地震波的传播时间中稳定地恢复这些系数。首席研究员将研究由远程观测引起的相对论中运动介质和洛伦兹度量的恢复问题。在医学成像中出现的断层扫描问题也将被研究。最后，主要研究者将研究线性和非线性算子的最佳采样率及其在反问题中的应用问题。现实生活中的测量是离散的，它们通常在小型探测器上平均，并且数值模拟也是在离散网格上完成的。首席研究员计划对各向同性弹性中弹性压力波和剪切波（奇点）的传播、反射、透射和模式转换进行全面分析，并在光滑表面上跳跃，模拟地幔之间的边界等。其中的一些分析已经在平坦常系数情况下做过了。首席研究员也将研究瑞利面波和斯通利面波。接下来，首席研究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。这个问题比黎曼问题更难，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（包括康普顿相机成像）中出现的断层扫描问题将被视为傅里叶积分算子，其分析将需要该微积分中的特定工具。最后，首席研究员计划研究线性和非线性算子的图像采样，这是出于测量离散数据能力的实际考虑。首席研究员将把采样理论与半经典分析联系起来，而不是研究“带限”（傅里叶变换的支持），他将把采样要求与半经典波前集联系起来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.