Iterative methods for Non-Hermitian Problems and Related Matrix Analysis

研究者Marko Huhtanen研究迭代方法并开发相关矩阵分析，用于解决线性代数中的大规模问题。重点是涉及非厄米矩阵的问题。最近，他将厄米矩阵的最优方法推广到普通矩阵问题。这就产生了两个根本不同的算法族。一种是基于经典厄米朗佐算法的直接扩展，另一种是基于实际的解析技术。这些方法可以用于求解线性系统和寻找特征值，以及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，研究者的目标是寻找将这些解决技术扩展到非正常问题的方法。研究的矩阵分析部分处理，例如，矩阵接近问题和矩阵类，矩阵向量乘积可以廉价地进行，通常使用FFT技术。这个项目背后的动机是提高解决现实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。那么要解决的未知数的数量可能是数百万。要在合理的时间限制内解决如此规模的问题，需要新的算法和求解技术。信号处理可以给出一个具体的例子：更快的算法意味着更快的信号处理，快速傅里叶变换已经彻底改变了这个特定的工程领域。研究者在数值分析的最基本水平上研究方法，因为线性系统需要在人们可以想象的每一个问题中得到实际解决。因此，这项研究的影响可能非常大。除了发明快速解决方法外，在这个项目中，开发了从纯矩阵分析的角度来看感兴趣的数学工具。

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.