Geometrical algorithms for the inverse scattering of waves

拟议的研究计划是一个跨学科的努力，旨在实现精确的计算解决方案，涉及复杂介质中弹性波和电磁波的体积散射的逆问题。例如，地震学就非常需要新的算法。目前首选的数值方法-有限差分格式，不参考波的基本几何形状-有一个严重的可扩展性问题，并没有给出任何提示，如何解决逆问题的硬非线性。我们认为，从微观和调和分析的见解建议转移大部分的计算负担，以预处理预测波运动学：1）通过传递到一个数值表示在相空间的运营商有关的波和亥姆霍兹方程，算法的正问题，可以开发恢复近线性复杂的波场数据; 2）直接在相位上拟合仿射-空间的最佳运输的想法，预计解决一些非凸性问题，否则会出现的反问题时，使用连续线性化。摩尔定律的指数增长的计算性能往往不匹配的指数进展，在计算科学。罪魁祸首是主流算法缺乏可扩展性：可以解决的问题规模增长比硬件能力增长更慢。在越来越多的应用中，需要数学家的投入来帮助工程师和应用科学家重新思考数字代码的设计，以避免这种可扩展性的诅咒。这个建议是一个努力后退一步，并介绍新的算法思想的地震成像，学科有关成像的地下地球。地震成像是能源部门用于碳氢化合物、水和地热能勘探的主要预测工具。它是水库监测技术和碳固存实验的核心。它被证明对那些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。所有这些远程成像问题现在已经成为我们这一代人将负责解决的极其复杂的计算问题。

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.