Integration of Randomized Methods and Fast and Reliable Matrix Computations

在现代科学计算、工程模拟和数据分析中，数值问题的复杂性和数据规模对矩阵计算提出了独特的挑战。对效率和可靠性的需求持续增长，与此同时，研究人员越来越希望使用方便的算法。随机算法不仅方便快捷，而且功能强大，因为它们有时可以从具有挑战性的情况中提取出令人惊讶的有用信息，否则很难处理。该项目将整合各种各样的随机化技术和一系列新颖的矩阵算法，以建立一个快速，可靠和灵活的随机化矩阵计算的综合框架。该项目将弥合方便的随机想法和各种具有挑战性的数字任务之间的差距。创新的随机方法将被设计为在数值计算中提取有价值的信息，并指导算法设计和参数调整。该项目将有助于有效地使更广泛的科学界更广泛地使用随机策略。它将有助于将新颖的随机算法引入各种数值分析领域，还可以显着提高数据科学，图像处理，地球科学和工程等应用领域中许多实际计算任务的效率和可靠性。研究成果将通过多种渠道广泛传播。该项目将为学生提供一个很好的平台，以获得不同领域的知识，如数值分析，统计和数据分析。将编制相关的课程材料。将设计开源软件包，研究将无缝集成各种随机技术和一系列新颖的矩阵算法。该研究将产生一系列新的随机计算方法，如低秩近似，数据稀疏预处理，特征值求解。将给出严格的理论来理解所提出的方法的有效性，并建立随机算法和各种具有挑战性的数值任务之间的联系。与许多随机化策略中效率和可靠性之间的通常折衷不同，多个随机化技术和新颖的矩阵计算的集成可以实现灵活性、方便性、效率和可靠性的组合益处。这些研究可以进一步帮助揭示内在矩阵特性，这些特性可用于设计鲁棒的数值算法来处理困难的情况。新的分析和随机方法将有助于使矩阵计算更好地满足现代计算任务的快速出现的挑战。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

In modern scientific computing, engineering simulations, and data analysis, the complexity of numerical problems and the scale of data sizes pose unique challenges to matrix computations. The demand for efficiency and reliability continues to grow, and in the meantime, researchers increasingly desire algorithms that are convenient to use. Randomized algorithms are not only convenient and fast to apply, but also powerful in the sense that they can sometimes extract surprisingly useful information from challenging situations that are otherwise very difficult to handle. This project will integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms so as to build a comprehensive framework for fast, reliable, and flexible randomized matrix computations. The project will bridge the gap between convenient randomized ideas and various challenging numerical tasks. Innovative randomized methods will be designed to extract valuable information in numerical computations and also to guide algorithm design and parameter tuning. The project will help effectively make randomized strategies more widely accessible to broader scientific communities. It will help introduce novel randomized algorithms into various numerical analysis fields and can also significantly improve the efficiency and reliability of many practical computational tasks in application fields such as data science, image processing, geosciences, and engineering. The research results will be widely disseminated via multiple channels. The project will give students a nice platform to gain knowledge in different areas such as numerical analysis, statistics, and data analysis. Relevant course materials will be developed. Open-source software packages will be designed.The research will seamlessly integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms. The research will result in a series of novel randomized methods for computations such as low-rank approximation, data-sparse preconditioning, and eigenvalue solution. Rigorous theories will be given to understand the effectiveness of the proposed methods and also to establish connections between randomized algorithms and various challenging numerical tasks. Unlike the usual compromise between efficiency and reliability in many randomized strategies, the integration of multiple randomized techniques and novel matrix computations can achieve the combined benefits of flexibility, convenience, efficiency, and also reliability. The studies can further help uncover intrinsic matrix properties that can be used to design robust numerical algorithms for handling difficult situations. The new analysis and randomized methods will help make matrix computations better meet the rapidly emerging challenges of modern computational tasks.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.