New numerical methods for Hamilton-Jacobi equations, Gaussian beams, and kinetic inverse problems

这位研究人员与他的学生和合作者一起，为哈密顿-雅可比方程、高斯光束和动力学逆问题开发了新颖而有效的数值方法。哈密顿-雅可比方程源于地震波传播、几何光学、最优控制、旅行时层析成像、医学成像、计算机视觉和材料科学。他以前的工作范围从三角网格上的哈密顿-雅可比方程的快速扫描方法，基于水平集的欧拉几何光学，到旅行时层析的快速数值方法。这些成功的工作使他为这些方程开发了更强大的数值方法，并将这些新的数值方法结合到地震建模和反演以及其他可能的应用中。所考虑的问题包括：发展基于勒让德变换的三角网格上定常哈密顿-雅可比方程的快速扫描方法，发展基于离散三角网格上的伊科诺方程的动力学逆问题的快速算法，发展动力学逆问题中测地线X射线变换的快速算法，发展高频波的欧拉高斯束，以及发展半经典量子力学的欧拉高斯束方法。这项研究促进了哈密顿-雅可比方程数值方法的发展，这些领域和应用在美国石油工业、医学成像以及材料科学和纳米技术中具有重要的战略价值。目前原油和其他地球资源价格的飙升越来越需要勘探地震学中更好的成像技术。全球地震学和勘探地震学中不断增长的数据量需要更复杂的数学模型。作为该项目的一部分开发的技术将为开发下一代地震成像工具提供关键工具，使地震勘探大大节省成本，并加快常规数据处理。

The investigator, with his students and collaborators, develops novel and efficient numerical methods for Hamilton-Jacobi equations, Gaussian beams, and kinetic inverse problems. Hamilton-Jacobi equations arise from seismic wave propagation, geometrical optics, optimal control, traveltime tomography, medical imaging, computer vision, and material sciences. His previous works range from fast sweeping methods for Hamilton-Jacobi equations on triangulated meshes, level-set based Eulerian geometrical optics, to fast numerical methods for traveltime tomography. These successful works lead him to develop more powerful numerical methods for these equations and incorporate these new numerical methods into seismic modeling and inversion, as well as other possible applications.Problems under consideration include developing Legendre-transform based fast sweeping methods for stationary Hamilton-Jacobi equations on triangulated meshes, developing fast algorithms for kinetic inverse problems based on discretizing eikonal equations on triangulated meshes, developing fast algorithms for geodesic X-ray transforms in kinetic inverse problems, developing Eulerian Gaussian beams for high frequency waves, and developing Eulerian Gaussian beam methods for semi-classical quantum mechanics.This investigation advances the state-of-the-art in numerical methods for Hamilton-Jacobi equations, high frequency wave propagation and kinetic inverse problems.These fields and applications are of great strategic value in the US petroleum industry, in the medical imaging, and in material sciences and nanotechnology. The current surge in price for crude oil and other earth resources increasingly demands better imaging techniques in exploration seismology. The increasing amount of data in global and exploration seismology requires more sophisticated mathematical models. The techniques developed as part of this project will provide crucial tools for the development of the next-generation seismic imaging tools that enable substantial cost savings in seismic explorations and expedite routine data processing.