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。实际的计算通常涉及大量数据，这是由于密集的矩阵或密集的中间矩阵块，这使经典矩阵方法不切实际。分层结构化方法提供了一种有效且可靠的方法来压缩和处理大型矩阵数据。在这种方法中，密集的矩阵块通过便利的处理方式近似。该研究项目旨在开发理论基础，以理解多个层次结构化技术并设计新的分层结构化算法。这些算法有望适用于更通用的矩阵计算和具有挑战性的应用，在这些应用中，通常的结构化方法不合适或有效。层次结构化方法利用矩阵计算中的固有结构，以提高效率高效率，同时确保出色的稳定性。该项目涉及快速可靠的层次结构化方法的设计，分析和应用，用于广泛的具有挑战性的计算类别。将提供一个统一的框架，以了解多种类型的层次结构化方法，设计具有增强适用性的新层次结构方法，并分析其准确性和稳定性。将开发最先进的快速和稳定求解器，以应对诸如大数据尺寸，疾病疾病，高频和多个频率之类的挑战。新求解器将适用于广泛的矩阵计算。基于数据稀疏性和增强稳定性，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。研究还将使快速稳定结构化的求解器可广泛地访问更广泛的领域和行业。数据将包含在数据存储库中，以进行无限制访问。开源套餐和教育/教程材料将免费提供。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，以紧密互动，并从多个领域学习批判性的计算和数学技能。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子的评估来支持的。和更广泛的影响审查标准。