Iterative methods for Non-Hermitian Problems and Related Matrix Analysis

非厄米问题的迭代方法及相关矩阵分析

• 批准号: 0209437
• 负责人: Alan Edelman
• 金额: $ 7.6万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0209437&HistoricalAwards=false
• 关键词: Iterative; methods; Non; Hermitian; Problems

Edelman 0209437 The investigator, Marko Huhtanen, studies iterative methods and develops related matrix analysis for solving large-scale problems in linear algebra. The emphasis is on problems involving non-Hermitian matrices. Recently he has extended optimal methods for Hermitian matrices to problems with normal matrices. This gave rise to two fundamentally different families of algorithms. One is based on a direct extension of the classical Hermitian Lanczos algorithm and the other on using real analytic techniques. These methods can be employed, e.g., in solving linear systems and finding eigenvalues, as well as in multivariate least squares approximation problems and interpolation. Moreover, he has introduced a classification of normal matrices to measure complexity of the algorithms. In light of these methods the investigator aims at finding ways to extend these solution techniques to nonnormal problems. The matrix analytic part of the study deals with, e.g., matrix nearness problems and classes of matrices with which matrix-vector products can be performed inexpensively, typically with the FFT techniques. The motivation behind the project is to increase the speed of computations for solving real world problems. Modeling problems of science and engineering realistically leads invariably to large scale problems. Then the number of unknowns to be solved can be millions. To solve problems of this size within a reasonable time limit calls for new algorithms and solution techniques. A concrete example can be given with signal processing: a faster algorithm means faster signal processing and the fast Fourier transformation has revolutionized this particular field of engineering. The investigator studies methods at the most fundamental level of numerical analysis because linear systems need to be solved practically in every problem one can imagine. Consequently, the impact of the study can be very large. In addition to inventing fast solution methods, in this project mathematical tools are developed that are of interest from a pure matrix analytic point of view.

Edelman 0209437 研究员Marko Huhtanen研究迭代方法，并开发了相关的矩阵分析来解决线性代数中的大规模问题。重点是涉及非埃尔米特矩阵的问题。最近，他扩大了最佳方法厄米特矩阵的问题与正常矩阵。这就产生了两个根本不同的算法家族。一个是基于直接扩展的经典埃尔米特Lanczos算法和其他使用真实的分析技术。这些方法可以被采用，例如，在解决线性系统和寻找特征值，以及在多元最小二乘近似问题和插值。此外，他还引入了正规矩阵的分类来衡量算法的复杂性。根据这些方法，调查员的目的是找到方法来扩展这些解决方案的技术，以非正常的问题。该研究的矩阵分析部分涉及，例如，矩阵近似问题和矩阵类，利用这些矩阵类，可以廉价地执行矩阵-向量乘积，通常利用FFT技术。 该项目背后的动机是提高解决真实的世界问题的计算速度。科学和工程的建模问题实际上总是导致大规模的问题。那么需要解决的未知数可能会达到数百万。要在合理的时间内解决这种规模的问题，需要新的算法和解决技术。一个具体的例子可以给出与信号处理：更快的算法意味着更快的信号处理和快速傅立叶变换已经彻底改变了这个特定的工程领域。研究人员研究的方法在最基本的水平的数值分析，因为线性系统需要解决实际上在每一个问题，人们可以想象。因此，这项研究的影响可能非常大。除了发明快速求解方法外，在这个项目中，还开发了从纯矩阵分析的角度来看感兴趣的数学工具。