Fast and Reliable Hierarchical Structured Methods for More General Matrix Computations

大型矩阵计算在现代科学计算任务和工程模拟中起着至关重要的作用。实际计算通常涉及大量的数据，由于大的密集矩阵或密集的中间矩阵块，这使得经典的矩阵方法不切实际。层次结构化方法为压缩和处理大型矩阵数据提供了一种有效可靠的方法。在这样的方法中，稠密矩阵块由便于处理的紧凑结构形式来近似。本研究计画的目的是为了解多重阶层式结构化技术及设计新的阶层式结构化演算法发展理论基础。这些算法有望应用于更一般的矩阵计算和具有挑战性的应用，通常的结构化方法是不合适的或有效的。分层结构化方法利用矩阵计算中的固有结构，以获得高效率，同时确保上级稳定性。该项目关注的是设计、分析和应用快速可靠的分层结构方法，用于广泛类别的具有挑战性的计算。将提供一个统一的框架来理解多种类型的分层结构方法，设计新的分层方法，增强适用性，并分析其准确性和稳定性。将开发最先进的快速和稳定的求解器，以应对大数据量、病态、高频和多频等挑战。新的求解器将适用于广泛的矩阵计算。基于数据稀疏性和增强的稳定性，求解器将显着提高PDE解决方案，大数据分析，网络，机器学习，成像，地震建模，电磁学等许多计算的效率和可靠性，该研究还将使快速，稳定的结构化求解器广泛应用于更广泛的领域和行业。这些数据将被纳入数据储存库，供不受限制地查阅。开放源码软件包和教育/辅导材料将免费提供。该多学科项目将为来自不同背景的研究生和本科生提供良好的机会，让他们密切互动，学习多个领域的关键计算和数学技能。该奖项反映了NSF的法定使命，并被认为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来支持。

Large matrix computations play a critical role in modern scientific computing tasks and engineering simulations. Realistic computations usually involve enormous amounts of data due to large dense matrices or dense intermediate matrix blocks, which makes classical matrix methods impractical. Hierarchical structured methods provide an effective and reliable way to compress and process large matrix data. In such methods, dense matrix blocks are approximated by compact structured forms that are convenient to handle. This research project aims to develop theoretical foundations for understanding multiple hierarchical structured techniques and for designing new hierarchical structured algorithms. These algorithms are expected to be applicable to more general matrix computations and challenging applications where usual structured methods are not suitable or effective.Hierarchical structured methods exploit inherent structures in matrix computations to gain high efficiency while ensuring superior stability. This project is concerned with the design, analysis, and application of fast and reliable hierarchical structured methods for broad classes of challenging computations. A unified framework will be provided to understand multiple types of hierarchical structured methods, design new hierarchical methods with enhanced applicability, and analyze their accuracy and stability. State-of-the-art fast and stable solvers will be developed for tackling challenges such as large data sizes, ill conditioning, high frequencies, and multiple frequencies. The new solvers will be applicable to a wide range of matrix computations. Based on data sparsity and enhanced stability, the solvers will significantly improve the efficiency and reliability of many computations in PDE solution, large data analysis, network, machine learning, imaging, seismic modeling, electromagnetics, etc. The research will also make fast and stable structured solvers widely accessible to broader fields and industries. The data will be included in data repositories for unrestricted access. Open-source packages and educational/tutorial materials will be freely available. The multidisciplinary project will provide excellent opportunities for graduate and undergraduate students from diverse backgrounds to closely interact and to learn critical computational and mathematical skills from multiple fields.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.