New Numerical Methods for Hamilton-Jacobi and Liouville Equations; Their Applications to Geometrical Optics, Wave Propagation and Travel-time Tomography

Hamilton-Jacobi 和 Liouville 方程的新数值方法；

• 批准号: 0510134
• 负责人: Jianliang Qian
• 金额: $ 9.1万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2005-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0510134&HistoricalAwards=false
• 关键词: New; Numerical; Methods; Hamilton; Jacobi

The investigator develops novel and efficient numericalmethods for Hamilton-Jacobi and Liouville equations. These equations arisefrom seismic wave propagation, geometrical optics, optimal control,travel-time tomography, medical imaging, computer vision, and materialsciences. His previous works on computing viscosity solutions andmultivalued solutions of Hamilton-Jacobi equations lead him to developmore powerful and efficient numerical methodsfor solving these equations and incorporate these new methodsinto seismic modeling and inversion, as well as other possibleapplications. Problems under consideration include continued developmentof adaptive eikonal solvers for three-dimensional isotropic andanisotropic media; fast sweeping methods for stationary Hamilton-Jacobiequations on unstructured meshes; extending slowness matching methods togeneral Hamilton-Jacobi equations for computing multivalued solutions; andapplications of paraxial Liouville equations for geometrical optics, wavepropagation and transmission tomography.This investigation advances the state-of-the-art in geometrical optics,wave propagation and seismic travel-time tomography.These fields and applications are of great strategic value in the US oiland gas industry, in environmental sciences, and in medical imaging.Recently, for example, the price of gasoline soared dramatically in theUnited States. One way to reduce the cost of production of oil involveslowering drilling costs by advancing the seismic data processingtechniques that oil companies use to find good drilling sites. Theinvestigator's new methods expedite routine data processing, provide newtools for exploration geophysicists to use for ground-breakingapplications, and enable substantial cost savings in seismic explorations,as the speed and reliability of the underlying computational engine allows``rig-site'' adjustments to both seismic survey and drilling decisions.

研究人员为 Hamilton-Jacobi 和 Liouville 方程开发了新颖且有效的数值方法。这些方程源自地震波传播、几何光学、最优控制、行进时间断层扫描、医学成像、计算机视觉和材料科学。他之前在计算 Hamilton-Jacobi 方程的粘度解和多值解方面的工作使他开发出更强大、更高效的数值方法来求解这些方程，并将这些新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继续开发三维各向同性和各向异性介质的自适应程算解算器；非结构化网格上平稳 Hamilton-Jacobie 方程的快速扫描方法；将慢度匹配方法扩展到通用 Hamilton-Jacobi 方程以计算多值解；以及近轴刘维尔方程在几何光学、波传播和透射层析成像中的应用。这项研究推进了几何光学、波传播和地震走时层析成像的最新技术。这些领域和应用在美国具有重大战略价值石油和天然气工业、环境科学和医学成像。例如，最近美国汽油价格大幅飙升。降低石油生产成本的一种方法是通过改进石油公司用来寻找良好钻井地点的地震数据处理技术来降低钻井成本。 研究人员的新方法加快了常规数据处理速度，为勘探地球物理学家提供了用于突破性应用的新工具，并在地震勘探中节省了大量成本，因为底层计算引擎的速度和可靠性允许对两者进行“钻机现场”调整地震勘探和钻井决策。