Geometrical algorithms for the inverse scattering of waves

提出的研究计划是一项跨学科的努力，旨在实现涉及复杂介质中弹性和电磁波体积散射的逆问题的数学知识计算解决方案。例如，地震学对新算法的需求很大。目前首选的数值方法——有限差分格式，没有参考波的潜在几何形状——有一个严重的可扩展性问题，并且没有提示如何解决逆问题的硬非线性。我们认为，来自微局部和谐波分析的见解建议将大部分计算负担转移到波运动学的预处理预测上：1)通过传递与波和亥姆霍兹方程相关的算子在相空间中的数值表示，可以开发出用于正演问题的算法来恢复波场数据的近线性复杂性；2)预测用最优输运思想直接在相空间拟合微分同态可以解决一些非凸性问题，否则在使用连续线性化求解逆问题时会出现。计算性能呈指数增长的摩尔定律往往与计算科学的指数进步不相匹配。罪魁祸首是主流算法缺乏可扩展性：可以解决的问题规模的增长速度比硬件能力的增长速度要慢。在越来越多的应用中，需要数学家的输入来帮助工程师和应用科学家重新思考数字代码的设计，以避免可扩展性的诅咒。这一提议是一项努力，旨在退后一步，为地震成像引入新的算法思想，地震成像是一门与地球地下成像有关的学科。地震成像是能源领域油气、水和地热能勘探的主要预测工具。它是水库和碳封存实验监测技术的核心。事实证明，它对争论地幔地质组成的地球物理学家很有用。高分辨率地震成像也开始使陆军和空军能够探测简易爆炸装置。到目前为止，所有这些远程成像问题都变成了极其复杂的计算问题，我们这一代人将负责解决这些问题。

The proposed research program is an interdisciplinary effort, aimed at implementing mathematically-informed computational solutions to inverse problems that involve volume scattering of elastic and electromagnetic waves in complex media. Seismology, for instance, is in great demand of new algorithms. The current preferred numerical methods--finite difference schemes that make no reference to the underlying geometry of waves--have a serious scalability issue, and give no hint on how to resolve the hard nonlinearity of the inverse problem. We argue that insights from microlocal and harmonic analysis suggest shifting much of the computational burden to a pre-processing predictive of wave kinematics: 1) by passing to a numerical representation in phase space for the operators related to the wave and Helmholtz equations, algorithms for the forward problem can be developed to restore near-linear complexity in the wave field data; and 2) fitting diffeomorphisms directly in phase-space by optimal transport ideas is predicted to resolve some of the nonconvexity issues otherwise arising when the inverse problem is solved using successive linearizations.Moore's law of exponential increase in computing performance is not often matched by exponential progress in the computational sciences. The culprit is the lack of scalability of mainstream algorithms: the size of problems that can be solved grows more slowly than hardware capabilities. In increasingly many applications, the input of mathematicians is needed to help engineers and applied scientists rethink the design of numerical codes to avoid this curse of scalability. This proposal is an effort to take a step back and introduce new algorithmic ideas for seismic imaging, the discipline concerned with imaging the subsurface of the Earth. Seismic imaging is the energy sector's main predictive tool for hydrocarbon, water, and geothermal energy prospection. It is at the heart of monitoring techniques for reservoirs and carbon sequestration experiments. It has proved useful to geophysicists who debate the geological composition of the Earth's mantle. High-resolution seismic imaging is also starting to enable the Army and the Air Force to detect IEDs. All these remote imaging problems have by now become formidably complex computational questions that our generation will be responsible for solving.