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，例如微分方程或差分方程的稳定性问题，或者求解线性系统的迭代方法的收敛性问题，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。在真实的对称矩阵（或者更一般的正规矩阵）的情况下，这些问题可以用矩阵的特征值来回答，但是当矩阵是高度非正规的（这意味着它的特征向量几乎是线性相关的或者它没有一个完整的特征向量集）时，特征值只能说明一部分问题。对于大指数，矩阵指数的幂或矩阵本身的渐近行为仍然由特征值决定，但瞬态行为不是。在这个项目中，主要研究者将使用复平面中可以与矩阵相关联的其他集合，例如值域或ε-伪谱，以提供有关此类矩阵函数行为的更多信息，例如有限幂。在另一个方向上，关于此类矩阵函数行为的新结果将导致对复逼近理论问题的新见解。该奖项将支持矩阵理论和复分析等数学领域的工作，这些工作在科学和工程的许多应用领域都是基础。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，飞机中的颤振最终会消失），但更关键的问题可能是近期会发生什么。虽然计算机模拟可以回答这个问题的一个特定的场景，一般的结果是什么决定了这样的系统的瞬态行为往往缺乏。这个项目将提供更好的数学工具来分析这类问题的广泛的模型。该奖项将通过研究助学金支持这些领域的研究生培训。