Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for high-frequency waves and inverse problems

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The investigator, with his students and collaborators, develops novel and efficient numerical methods for modeling high-frequency wave propagation and solution of associated inverse problems. These problems arise from seismic wave propagation, geometrical optics, optimal control, computerized tomography (CT), medical imaging, and material sciences. The proposal focuses on advancing fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods, a novel approach for this challenging problem, from theoretical, algorithmic, and practical perspectives. This interdisciplinary research complements the PI's educational goals by integrating education and research activities at undergraduate and graduate levels. Problems under consideration include exploring the deep connection between multiscale Gaussian beams and fast multiscale Gaussian wavepacket transforms to devise new algorithms for decomposing given data into Gaussian beams, developing new multiscale Gaussian beam methods for modeling acoustic, elastic and anisotropic waves, analyzing linearized inverse problems for acoustic and elastic wave equations by using multiscale Gaussian beams in the high frequency regime, devising novel algorithms to implement the resulting linearized inversion formulas, and validating the resulting algorithms by using synthetic data.Modeling of high frequency waves is of great strategic value in diverse science and engineering disciplines, ranging from the US petroleum industry, seismic imaging, radar, sonar, medical imaging, remote sensing, submarine detection, material sciences to 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 provide crucial tools for the development of the next-generation seismic imaging tools with the potential to enable substantial cost savings in seismic explorations, expedite routine data processing, and protect the environment by optimizing drilling sites. Students from the PI's institution are involved in this innovative interdisciplinary research project.