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

Hamilton-Jacobi 方程、高斯梁和动力学反问题的新数值方法

• 批准号: 0810104
• 负责人: Jianliang Qian
• 金额: $ 17.06万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810104&HistoricalAwards=false
• 关键词: New; numerical; methods; Hamilton; Jacobi

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.

研究人员与他的学生和合作者一起，为哈密尔顿-雅可比方程、高斯梁和动力学反问题开发了新颖且有效的数值方法。汉密尔顿-雅可比方程源自地震波传播、几何光学、最优控制、走时断层扫描、医学成像、计算机视觉和材料科学。他之前的工作范围包括三角网格上的 Hamilton-Jacobi 方程的快速扫描方法、基于水平集的欧拉几何光学，以及走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中。正在考虑的问题包括开发基于勒让德变换的静止汉密尔顿-雅可比快速扫描方法三角网格上的方程，基于三角网格上的离散化函方程开发动力学逆问题的快速算法，开发动力学逆问题中的测地X射线变换的快速算法，开发欧拉用于高频波的高斯光束，并开发用于半经典量子力学的欧拉高斯光束方法。这项研究推进了 Hamilton-Jacobi 方程、高频波传播和动力学反问题的数值方法的最新技术。这些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大战略价值。当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。全球和勘探地震学数据量的不断增加需要更复杂的数学模型。作为该项目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，从而大幅节省地震勘探成本并加快常规数据处理速度。