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Fast and Reliable Hierarchical Structured Methods for More General Matrix Computations

用于更一般矩阵计算的快速可靠的分层结构化方法

基本信息

批准号：
1819166
负责人：
Jianlin Xia
金额：
$ 19.91万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819166&HistoricalAwards=false
关键词：
Fast Reliable Hierarchical Structured Methods

项目摘要

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.

大矩阵计算在现代科学计算任务和工程仿真中起着至关重要的作用。现实计算中，由于存在较大的密集矩阵或密集的中间矩阵块，通常会涉及到大量的数据，这使得经典的矩阵方法变得不切实际。分层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。在这种方法中，密集矩阵块由易于处理的紧凑结构形式近似。本研究项目旨在为理解多层分层结构技术和设计新的分层结构算法奠定理论基础。这些算法有望适用于更一般的矩阵计算和具有挑战性的应用，通常的结构化方法不适合或有效。层次结构方法利用矩阵计算的固有结构，在保证良好稳定性的同时获得高效率。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，用于各种具有挑战性的计算。将提供一个统一的框架来理解多种类型的分层结构方法，设计新的具有增强适用性的分层方法，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，以应对诸如大数据规模、病态、高频和多频率等挑战。新的求解器将适用于广泛的矩阵计算。基于数据的稀疏性和增强的稳定性，求解器将显著提高PDE求解、大数据分析、网络、机器学习、成像、地震建模、电磁学等领域许多计算的效率和可靠性。这项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。数据将包含在数据存储库中，以便不受限制地访问。开源软件包和教育/教程材料将免费提供。这个多学科项目将为来自不同背景的研究生和本科生提供绝佳的机会，让他们密切互动，学习多个领域的关键计算和数学技能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。