CDI-Type I: Collaborative Research: High-Dimensional Phase-Space Subdivisions for Seismic Imaging

该奖项支持一项研究计划，该计划旨在设计数学知识丰富的计算工具，用于处理沿低维流形显示定向结构的大型高维地震数据集。过去几十年在地震成像方面取得的进展在很大程度上忽略了与数据相关的日益增长的复杂性，如相干噪声、多次散射、不规则采集几何和同时采集。计算调和分析通过构造利用变换域中的稀疏性的优化问题来提供这些问题的解决方案。然而，这些工具不能用于非常大规模的反演任务，因为它们在这种情况下在计算上没有优势。这个项目回顾了多尺度定向变换的数学基础，以期设计出即使是最密集的数据密集型反转场景也应该具有竞争力的低冗余、高维体系结构。计算性能的指数增长的摩尔定律通常不会与计算科学的指数进步相匹配。罪魁祸首是主流算法缺乏可扩展性：可以解决的问题的规模比硬件能力增长得更慢。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考虑数字代码的设计，以避免这种可伸缩性的诅咒。这个项目是为了退后一步，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。地震成像是能源部门对碳氢化合物、水和地热能勘探的主要预测工具。它是水库监测技术和碳封存实验的核心。事实证明，它对就地幔的地质组成进行辩论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到简易爆炸装置。到目前为止，所有这些远程成像问题都已经变成了极其复杂的计算问题，我们这一代人将负责解决。

This award supports a research program aimed at designing mathematically-informed computational tools for processing large, high-dimensional seismic datasets that display directional structure along lower-dimensional manifolds. The progress that occurred over the past few decades in seismic imaging has largely ignored growing data-related complications, such as coherent noise, multiple scattering, irregular acquisition geometries, and simultaneous acquisition. Computational harmonic analysis provides solutions to these problems by formulating optimization problems that leverage sparsity in a transformed domain. These tools can however not be relied upon for very large scale inversion tasks, because they are not computationally advantageous in such regimes. This project revisits the mathematical underpinnings of multiscale directional transforms with a view toward designing low-redundancy, high-dimensional architectures that should be competitive for even the most data-intensive inversion scenarios.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 project 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.