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

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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.