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Generalized Matrix Functions: Theory, Algorithms, and Applications

广义矩阵函数：理论、算法和应用

基本信息

批准号：
1719578
负责人：
Michele Benzi
金额：
$ 25万
依托单位：
Emory University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719578&HistoricalAwards=false
关键词：
Generalized Matrix Functions Theory Algorithms

项目摘要

In recent years the scientific enterprise, like daily existence, has been greatly affected by the availability of new technologies that have enabled the collection of an unprecedented amount of data. This continuous creation of enormous amounts of raw digital information demands new, efficient ways to extract useful content and filter out the noise inherent in any data-gathering process. Mathematical and computational techniques of data analysis offer powerful tools that can be brought to bear, but new challenges arise on a daily basis. This requires the constant refinement and improvement of existing techniques, as well as the development of new ones. This research project aims to produce a new body of mathematical knowledge and new computational techniques that will enhance the ability to tackle challenges arising from data-intensive fields of science and engineering including computer vision, compressed sensing, control, and others. Quantitative finance, network analysis and synthesis, and medical imaging are other areas where the research can be expected to have an impact. The principal investigator will study a class of mathematical objects known as generalized matrix functions and aims to exploit the resulting knowledge to develop new, efficient computer methods for data analysis. Training of a PhD student in computational mathematics is also an integral part of the project.The principal investigator aims to develop the theory of generalized matrix functions, a type of matrix function based on the singular value decomposition of a (possibly rectangular) matrix. The resulting theory is intended to be the basis for the development of numerical methods for the efficient approximate evaluation of such matrix functions. The focus will be primarily on large-scale problems for which the (full) singular value decomposition cannot be computed. Techniques based on sparsity and low-rank approximations will be combined with Krylov-type methods (especially the Lanczos and Golub--Kahan algorithms) to design fast algorithms for solving a variety of problems involving generalized matrix functions. The algorithms will be applied to problems such as low-rank matrix optimization, the regularization of discrete inverse problems, and the analysis of directed networks. As a by-product of this research, new algorithms for the computation of standard matrix functions where the matrix argument is only available in factored form will be derived and analyzed. This research represents a new direction in numerical linear algebra and is expected to produce useful numerical tools for the solution of a variety of problems in data science and optimization.

近年来，科学事业和日常生活一样，受到了新技术的极大影响，这些技术使收集到的数据数量达到了前所未有的水平。大量原始数字信息的不断产生需要新的、有效的方法来提取有用的内容，并过滤掉任何数据收集过程中固有的噪音。数据分析的数学和计算技术提供了强大的工具，但每天都有新的挑战出现。这需要不断改进和提高现有技术，并开发新技术。该研究项目旨在产生新的数学知识和新的计算技术，以提高应对科学和工程数据密集型领域（包括计算机视觉，压缩传感，控制等）挑战的能力。定量金融、网络分析和综合以及医学成像是预计研究会产生影响的其他领域。首席研究员将研究一类被称为广义矩阵函数的数学对象，旨在利用由此产生的知识开发新的，有效的计算机数据分析方法。培养一名计算数学博士生也是该项目的一个组成部分。主要研究者的目标是发展广义矩阵函数的理论，广义矩阵函数是一种基于（可能是矩形）矩阵奇异值分解的矩阵函数。由此产生的理论的目的是发展的数值方法的基础上，有效的近似评估等矩阵函数。重点将主要放在大规模的问题，其中（全）奇异值分解无法计算。基于稀疏性和低秩近似的技术将与Krylov型方法（特别是Lanczos和Golub-Kahan算法）相结合，以设计快速算法来解决涉及广义矩阵函数的各种问题。该算法将被应用于低秩矩阵优化，离散逆问题的正则化，以及有向网络的分析等问题。作为这项研究的副产品，新的算法计算的标准矩阵函数的矩阵参数是唯一的因式分解的形式将被导出和分析。这项研究代表了数值线性代数的一个新方向，预计将为解决数据科学和优化中的各种问题提供有用的数值工具。