Collaborative Research: Wave Computations in Phase-Space

负责人提案：DMS -0707921 PI： Demanet，Laurent Institution 斯坦福大学非牵头方案：DMS-0708014 PI： 影，乐行机构： 德克萨斯大学奥斯汀分校 合作研究：地震成像领域目前正面临着一个主要的计算挑战，因为反演算法的能力增长速度比采集数据量慢。反演的成功通常取决于在大规模计算规模上求解波动或双曲方程或其适当近似的实用性。为此，PI建议在二维和三维空间中重新审视光滑介质中的计算波传播，以便将复杂性降低到初始数据大小的渐近线性，达到对数因子和合理常数。在这种低复杂度的制度，涉及绿色的功能成为数值工作的主要焦点预先计算。为此，PI建议设计，实施，测试和分析以下数值方法：（1）傅立叶积分算子（FIO）的有效算法，使用诸如相空间划分、几何下采样、方向插值和经由随机采样的低秩矩阵近似的技术，（2）具有平滑系数的线性双曲PDE的有效算法，基于FIO的上述算法，并且还使用诸如用于传播时间的相流方法、伪微分符号的分离和随机采样以及利用波传播的微局部几何形状的特殊求积的技术，以及（3）基于FIO的上述算法的地震成像中的Kirchhoff偏移的有效算法，并且还涉及用于成像操作者的运动学的高维压缩技术。在一个单独的努力，PI将探索更一般的情况下，物理利益，如相位爆破和多路径，这将需要新的ideas.The拟议的研究是直接的动机，需要新的，有效的反演方法在反射地震。反过来，改进的地震成像技术（1）可以帮助发现新的物理学和解决地球物理学中现有的争论（例如关于地幔中的对流现象），以及（2）可以为工业勘探提供更好的地球上地壳地图。PI计划在项目的后期阶段与地震学家密切合作，以提供操作代码并在地球物理学界传播想法。另外，透射电子显微镜是另一个曲线断层成像的问题，所提出的算法将提供一个新的前景，对新的，准确的反演方法，在生物学和医学成像的应用。

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.