Collaborative Research: Wave Computations in Phase-Space

主要建议：DMS-0707921PI：Demanet，Laurent Institution：Stanford University非主要建议：DMS-0708014PI：Ying，Lexing Institution：德克萨斯大学奥斯汀分校标题：协作研究：相空间中的波计算摘要地震成像领域目前面临着重大的计算挑战，因为反演算法的能力增长速度慢于采集数据量。反演的成功通常取决于在大规模计算范围内解波动或双曲型方程或它们的适当近似的实用性。为此，PI建议重新讨论光滑介质中二维和三维空间中的计算波传播，以便将复杂性降低到初始数据大小的渐近线性，直到对数因子和合理常数。在这种低复杂性的情况下，涉及格林函数的预计算成为数值计算的主要焦点。为此，PI建议设计、实现、测试和分析以下数值方法：(1)有效的傅里叶积分算子(FIO)算法，使用相空间划分、几何下采样、定向内插和通过随机采样的低阶矩阵近似等技术；(2)基于上述FIO算法，并使用诸如走时的相流法、伪微分符号的分离和随机采样以及利用波传播的微观局部几何的特殊求积方法等技术，有效地求解地震成像中的Kirchhoff偏移，基于FIO的上述算法，以及基于高维压缩技术的成像算子的运动学。在另一项工作中，PI将探索更一般的物理感兴趣的情况，如相位爆炸和多路径，这将需要新的想法。拟议的研究直接受到反射地震学对新的、有效的反演方法的需求的推动。反过来，改进的地震成像技术(1)可以帮助发现新的物理现象并解决地球物理学中现有的争论(例如关于地幔对流现象)，以及(2)可以为工业勘探目的提供更好的地球上地壳地图。PIS计划在该项目的后期阶段与地震学家密切合作，在地球物理学界传递业务代码和传播思想。或者，透射式电子显微镜是另一个曲线层析成像问题，所提出的算法将为新的、准确的反演方法提供新的前景，并在生物学和医学成像中应用。

Lead Proposal: DMS - 0707921PI: Demanet, Laurent Institution: Stanford UniversityNon-Lead Proposal: DMS-0708014PI: Ying, Lexing Institution: University of Texas at AustinTitle: Collaborative Research: Wave Computations in Phase-SpaceABSTRACTThe field of seismic imaging is currently facing a major computational challenge, because the capabilities of inversion algorithms grow at a slower pace than the volume of acquired data. Success of inversion typically hinges on the practicality of solving wave or hyperbolic equations, or proper approximations thereof, on a massive computational scale. To this end, the PIs propose to revisit computational wave propagation in smooth media, in two and three space dimensions, in order to bring the complexity down to asymptotically linear in the size of the initial data, up to log factors and reasonable constants. In this low-complexity regime, precomputations involving the Green's function become the main focus of the numerical effort. To this end, the PIs propose to design, implement, test and analyze the following numerical methods: (1) an efficient algorithm for Fourier Integral Operators (FIO), using techniques such as phase-space partitionings, geometric downsamplings, directional interpolation, and low rank matrix approximations via random sampling, (2) an efficient algorithm for linear hyperbolic PDE with smooth coefficients, based on the above algorithm for FIO, and also using techniques such as the phase-flow method for travel times, separation and random samplings of pseudodifferential symbols, and specialquadratures that exploit the microlocal geometry of wave propagation, and (3) an efficient algorithm for Kirchhoff migration in seismic imaging, based on the above algorithm for FIO, and also on a high-dimensional compression technique for the kinematics of the imaging operator. In a separate effort, the PIs will explore more general situations of physical interest such as phase blowups and multipathing, for which new ideas will be required.The proposed research is directly motivated by the need for new, efficient inversion methods in reflection seismology. In turn, improved seismic imaging techniques (1) could help discover new physics and settle existing debates in geophysics (for instance concerning convection phenomena in the Earth's mantle), and (2) could provide a better map of the Earth's upper crust, for industrial exploration purposes. The PIs plan on working closely with seismologists in the later phases of the project, to deliver operational codes and disseminate ideas in the geophysics community. Alternatively, transmission electron microscopy is another curvilinear tomography imaging problem for which the proposed algorithms will provide a fresh outlook towards novel, accurate inversion methods, with applications in biology and medical imaging.