Inverse Problems in Partial Differential Equations and Geometry

该项目的主要重点是从遥感测量中确定介质参数的逆问题，特别是在地球探测、宇宙学和移动和变化介质成像、医学成像中产生的逆问题，以及线性和非线性逆问题的采样理论。 主要研究者将研究弹性地球模型，该模型的系数在光滑表面上跳跃，压力波和剪切波在其中的传播和模式转换。目标是证明人们可以从地震波的旅行时间稳定地恢复这些系数。 主要研究者将研究从远程观测中恢复运动介质和相对论中产生的洛伦兹度量的问题。 在医学成像中出现的断层摄影问题也将被研究。 最后，首席研究员将研究如何找到线性和非线性算子的最佳采样率，并将其应用于逆问题。 对那些感兴趣的动机是这样一个事实，即真实的生命测量是离散的，它们通常在小探测器上平均，并且也在离散网格上进行数值模拟。主要研究人员计划做一个全面的分析传播，反射，传输和模式转换的弹性压力和剪切波（奇点）在各向同性弹性与变系数跳跃光滑表面建模地幔之间的边界等，这种分析已经完成了一些平坦的常数系数的情况下。首席研究员将研究瑞利和斯通利波以及表面波。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和从远程观测恢复洛伦兹度量的非线性问题。这个问题比它的黎曼对应问题更难，因为比光速更快的信号是无法以稳定的方式恢复的。在医学成像中出现的层析成像问题，包括康普顿照相机成像，将被视为傅立叶积分算子，其分析将需要特定的工具，从微积分。最后，主要研究者计划研究线性和非线性算子的图像采样，其动机是仅测量离散数据的能力的实际考虑。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.