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

研究者，与他的学生和合作者，开发新的和有效的数值方法，汉密尔顿-雅可比方程，高斯光束，和动力学逆问题。Hamilton-Jacobi方程来源于地震波传播、几何光学、最优控制、旅行时断层扫描、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Hamilton-Jacobi方程的快速扫描方法，基于水平集的欧拉几何光学，到旅行时断层扫描的快速数值方法。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纳入地震建模和反演，以及其他可能的应用。考虑的问题包括：在三角网格上发展基于legende变换的平稳Hamilton-Jacobi方程快速扫描方法；在三角网格上发展基于离散化eikonal方程的运动反问题快速算法；在运动反问题中发展测地x射线变换快速算法；发展了半经典量子力学的欧拉高斯束方法。本研究在Hamilton-Jacobi方程、高频波传播和动力学反问题的数值方法上取得了新的进展。这些领域和应用在美国石油工业、医学成像、材料科学和纳米技术方面具有重要的战略价值。当前原油和其他地球资源价格的飙升日益要求在勘探地震学中采用更好的成像技术。全球和勘探地震学数据量的增加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，从而大大节省地震勘探成本，加快常规数据处理速度。

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.