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

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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.