Applied Matrix Theory and Complex Approximation: Estimating Norms of Functions of Matrices

应用矩阵理论和复近似：估计矩阵函数的范数

• 批准号: 1210886
• 负责人: Anne Greenbaum
• 金额: $ 35.29万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1210886&HistoricalAwards=false
• 关键词: Applied; Matrix; Theory; Complex; Approximation

Many questions in applied and computational mathematics, such as questions about the stability of differential or difference equations or questions about the convergence of iterative methods for solving linear systems, are really questions about norms of functions of matrices, such as matrix exponentials or matrix powers. In the case of a real symmetric matrix (or, more generally, a normal matrix), these questions are answered in terms of the matrix eigenvalues, but when the matrix is highly nonnormal (meaning that its eigenvectors are nearly linearly dependent or it does not have a complete set of eigenvectors), eigenvalues tell only part of the story. The asymptotic behavior of powers of matrix exponentials or the matrix itself for large exponents is still determined by the eigenvalues, but transient behavior is not. In this project, the principal investigator will use other sets in the complex plane that can be associated with a matrix, such as the field of values or the epsilon-pseudospectrum, to give more information about the behavior of such matrix functions e.g. for finite powers. In the other direction, new results about the behavior of such matrix functions will lead to new insights into problems in complex approximation theory.This award will support work in the mathematical areas of matrix theory and complex analysis that is fundamental in many application areas in science and engineering. A range of mathematical methods is available for describing the long-term behavior of systems (will a radiation level eventually decay to zero, will flutter in an aircraft eventually die out), but a more crucial question may be what happens in the near-term. While a computer simulation may answer this question for a specific scenario, general results about what determines the transient behavior of such systems are often lacking. This project will provide better mathematical tools for analyzing such questions for a wide class of models. The award will support graduate student training in these areas through research assistantships.

应用数学和计算数学中的许多问题，例如关于微分或差分方程的稳定性的问题或关于求解线性系统的迭代方法的收敛性的问题，实际上是关于矩阵函数的范数的问题，例如矩阵指数或矩阵幂。在实数对称矩阵（或更一般地，正态矩阵）的情况下，这些问题可以根据矩阵特征值来回答，但是当矩阵高度非正态时（意味着其特征向量几乎线性相关或不线性相关）有一套完整的特征向量），特征值只讲述了故事的一部分。矩阵指数幂或大指数矩阵本身的渐近行为仍然由特征值决定，但瞬态行为则不然。在这个项目中，主要研究者将使用复平面中可以与矩阵关联的其他集合（例如值域或 epsilon 伪谱）来提供有关此类矩阵函数行为的更多信息，例如对于有限的权力。另一方面，有关此类矩阵函数行为的新结果将带来对复数逼近理论问题的新见解。该奖项将支持矩阵理论和复数分析数学领域的工作，这在许多科学应用领域中都是基础和工程。有一系列数学方法可用于描述系统的长期行为（辐射水平最终会衰减到零，飞机上的颤振最终会消失），但更关键的问题可能是短期内会发生什么。虽然计算机模拟可以针对特定场景回答这个问题，但通常缺乏决定此类系统瞬态行为的一般结果。该项目将为分析各种模型的此类问题提供更好的数学工具。该奖项将通过研究助学金支持这些领域的研究生培训。