CAREER: Algorithms for Eigenvalue and Singular Value Problems

职业：特征值和奇异值问题的算法

• 批准号: 9702866
• 负责人: Ming Gu
• 金额: $ 20.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-03-01 至 2002-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9702866&HistoricalAwards=false
• 关键词: CAREER; Algorithms; Eigenvalue; Singular; Value

The research part of this CAREER project develops accurate and efficient numerical methods for a number of important problems including high relative accuracy (tiny percentage error) eigenvalue and singular value computations, updating the singular value decomposition (SVD), and distance problems in linear systems and control computations. Although current methods for solving them are abundant, many important questions remain unanswered. For example, a large number of recent papers describe numerical methods to compute all eigenvalues and singular values to high relative accuracy for a few isolated special classes of matrices, but a common explanation and a common numerical method are missing. As another example, the ability to rapidly update the SVD is critical in signal processing applications since problems are often real- time, but none of the current methods is efficient enough for this purpose. A goal is to provide a common explanation and a common numerical method for high relative accuracy eigenvalue and singular value computations. It turns out that the common explanation and common numerical method are applicable to many new interesting classes of matrices, arising from areas as diverse as combinatorics and numerical solution of ODE's and PDE's. The project also aims at providing rapid methods for updating the SVD, hence removing a major bottleneck in SVD-based methods for various signal processing applications. The education part of the project is to develop software tools so that students taking abstract mathematics courses can clearly envision and play with the sometimes abstract objects and results on the World-Wide Web with vivid graphical display. The plan also includes new graduate courses in numerical linear algebra with scientific and engineering applications.

该CAREER项目的研究部分为一些重要问题开发了准确有效的数值方法，包括高相对精度（微小百分比误差）特征值和奇异值计算，更新奇异值分解（SVD）以及线性系统和控制计算中的距离问题。 尽管目前解决这些问题的方法很多，但许多重要问题仍然没有得到解答。例如，最近的大量论文描述了计算所有特征值和奇异值的数值方法，以高相对精度为一些孤立的特殊类别的矩阵，但缺乏一个共同的解释和共同的数值方法。作为另一个例子，快速更新SVD的能力在信号处理应用中是关键的，因为问题通常是真实的-时间的，但是没有一个当前方法对于该目的是足够有效的。 一个目标是提供一个共同的解释和一个共同的数值方法，高相对精度的特征值和奇异值计算。事实证明，共同的解释和共同的数值方法适用于许多新的有趣的矩阵类，所产生的领域不同的组合和数值解的常微分方程和偏微分方程的。该项目还旨在提供快速更新SVD的方法，从而消除各种信号处理应用中基于SVD的方法的主要瓶颈。 该项目的教育部分是开发软件工具，使学生采取抽象的数学课程可以清楚地设想和发挥有时抽象的对象和结果在万维网上生动的图形显示。该计划还包括新的研究生课程，在数值线性代数与科学和工程应用。