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

研究者为Hamilton-Jacobi和Liouville方程发展了新颖有效的数值方法。这些方程来自地震波传播、几何光学、最优控制、走时层析成像、医学成像、计算机视觉和材料科学。他以前在计算粘性解和Hamilton-Jacobi方程的多值解方面的工作使他开发了更强大和有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。目前正在考虑的问题包括：继续发展三维各向同性和各向异性介质的自适应程函求解器;非结构网格上定常Hamilton-Jacobi方程的快速扫描方法;将慢度匹配方法推广到计算多值解的一般Hamilton-Jacobi方程;以及近轴刘维尔方程在几何光学中的应用，波传播和透射层析成像。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石油和天然气工业中具有重要的战略价值，例如，最近美国的汽油价格急剧上涨。降低石油生产成本的一种方法是通过提高石油公司用来寻找好的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，为勘探地球物理学家提供了新的工具，用于地面突破应用，并使地震勘探的成本大幅节省，因为底层计算引擎的速度和可靠性允许对地震勘探和钻井决策进行“钻机现场”调整。

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.