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

该研究员与他的学生和合作者一起开发了新颖有效的数值方法，用于模拟高频波传播和相关逆问题的解决方案。这些问题来自地震波传播、几何光学、最优控制、计算机断层扫描（CT）、医学成像和材料科学。该提案的重点是推进快速多尺度高斯波包变换和多尺度高斯束方法，这是从理论、算法和实践角度解决这一具有挑战性问题的一种新方法。这种跨学科的研究补充了PI的教育目标，通过整合本科和研究生水平的教育和研究活动。正在考虑的问题包括探索多尺度高斯波束和快速多尺度高斯波包变换之间的深层联系，以设计用于将给定数据分解为高斯波束的新算法，开发用于建模声波、弹性波和各向异性波的新的多尺度高斯波束方法，通过在高频区使用多尺度高斯波束来分析声波和弹性波方程的线性化逆问题，设计新的算法来实现所得到的线性化反演公式，并通过使用合成数据来验证所得到的算法。高频波的建模在各种科学和工程学科中具有重要的战略价值，从美国石油工业，地震成像，雷达，声纳，医学成像，遥感，潜艇探测，材料科学到纳米技术。目前原油和其他地球资源价格的飙升对勘探地震学中成像技术的要求越来越高。全球地震学和勘探地震学中数据量的增加需要更复杂的数学模型。作为该项目的一部分开发的技术为开发下一代地震成像工具提供了关键工具，这些工具有可能在地震勘探中节省大量成本，加快常规数据处理，并通过优化钻井地点来保护环境。PI机构的学生参与了这一创新的跨学科研究项目。

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.