The investigator develops novel and efficient numerical methods for Hamilton-Jacobi and Liouville equations. These equations arise from seismic wave propagation, geometrical optics, optimal control, travel-time tomography, medical imaging, computer vision, and material sciences. His previous works on computing viscosity solutions and multivalued solutions of Hamilton-Jacobi equations lead him to develop more powerful and efficient numerical methods for solving these equations and incorporate these new methods into seismic modeling and inversion, as well as other possible applications. Problems under consideration include continued development of adaptive eikonal solvers for three-dimensional isotropic and anisotropic media; fast sweeping methods for stationary Hamilton-Jacobi equations on unstructured meshes; extending slowness matching methods to general Hamilton-Jacobi equations for computing multivalued solutions; and applications of paraxial Liouville equations for geometrical optics, wave propagation 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 oil and gas industry, in environmental sciences, and in medical imaging. Recently, for example, the price of gasoline soared dramatically in the United States. One way to reduce the cost of production of oil involves lowering drilling costs by advancing the seismic data processing techniques that oil companies use to find good drilling sites. The investigator's new methods expedite routine data processing, provide new tools for exploration geophysicists to use for ground-breaking applications, 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.
