Matrix Functions, Rational Approximation, and Quadrature with Applications

这项研究项目致力于开发和分析矩阵函数和由矩阵函数定义的积分的逼近的新方法。这项研究融合了线性代数和逼近理论。关于厄米特矩阵的新的有理Lanczos方法正在发展中，这些方法具有较短的递归公式。短递归关系的存在具有重要的理论意义和实用价值。它加快了计算速度，因为每个新的有理Lanczos向量只需与最近计算的几个Lanczos向量进行正交化。所需的显式正交化的数目是有界的，与有理Lanczos步数无关。递归关系定义了给定矩阵在有理Krylov子空间上的正交投影。投影由五对角线矩阵表示，该矩阵也具有块结构。该矩阵类似于与(标准)Gauss求积规则相关的对称三对角矩阵，并且是计算有理Gauss、Gauss-Radau规则和Gauss-Lobatto求积规则的新算法的基础。这项研究是吉恩·戈卢布和他的合作者在矩阵、矩、正交多项式和求积方面非常重要的工作的扩展。可以连接到在单位圆上的正交多项式和正交有理函数的上下文中出现的结构化矩阵，例如CMV矩阵，以及到半可分和准可分矩阵。当给定的矩阵是非厄米特矩阵时，可以应用有理Arnoldi过程和有理非厄米特Lanczos过程。在有理Krylov子空间上的投影又有一个结构，它的利用是线性代数中的一个重要课题。估计矩阵泛函的需求出现在许多应用中，包括社会网络的研究和不适定反问题的Tikhonov正则化。主要研究人员正在研究更快的数值方法的发展，用于计算、界定或估计涉及大型对称或非对称矩阵的复杂非线性表达式。速度的提高是通过利用到目前为止一直被忽视的结构来实现的。在解决科学家和工程师感兴趣的大规模问题时，快速方法的开发是重要的。这些方法适用于对社交网络的调查，其性质最近受到了相当大的关注，不仅受到了科学家的关注，也受到了《纽约时报》的关注。调查社交网络的一个主要障碍是它们的庞大规模。这项研究为研究这类网络提供了改进的工具。该方法也可应用于三维麦克斯韦方程的求解。本质上，我们必须计算非常大的矩阵的指数。这些矩阵太大了，不允许使用标准软件。在这一建议框架内开发的方法通过利用这些问题的内在结构来加快计算速度。关于这个建议的问题的工作需要线性代数、正交多项式和有理函数、逼近理论和高斯型求积的背景知识。因此，这项研究非常适合从事该项目的博士生。

This research project is concerned with the development and analysis of novel methods for the approximation of matrix functions and integrals defined by matrix functions. The research blends linear algebra and approximation theory. New rational Lanczos methods for Hermitian matrices with short recursion formulas are being developed. The existence of short recursion relations is of significant theoretical and practical interest. It speeds up the computations because each new rational Lanczos vector only has to be orthogonalized against a few of the most recently computed Lanczos vectors. The number of required explicit orthogonalizations is bounded independently of the number of rational Lanczos steps. The recursion relations define the orthogonal projection of a given matrix onto a rational Krylov subspace. The projection is represented by a pentadiagonal matrix, which also has a block structure. This matrix is analogous to the symmetric tridiagonal matrix associated with a (standard) Gauss quadrature rule, and is the basis for new algorithms for the evaluation of rational Gauss, Gauss-Radau rules, and Gauss-Lobatto quadrature rules. The research is an extension of the very important work by Gene Golub, and his collaborators, on matrices, moments, orthogonal polynomials, and quadrature. There may be connections to the structured matrices, such as the CMV-matrices, which arise in the context of orthogonal polynomials and orthogonal rational functions on the unit circle, as well as to to semi- and quasi-separable matrices. When the given matrix is non-Hermitian, rational Arnoldi and rational non-Hermitian Lanczos processes can be applied. The projections onto the rational Krylov subspace again have a structure, the exploitation of which is an important topic in linear algebra. The need to estimate matrix functionals arises in many applications, including the investigation of social networks and in Tikhonov regularization of ill-posed inverse problems. The principal investigator is studying developments of faster numerical methods for the evaluation, bounding or estimation of complicated nonlinear expressions that involve large symmetric or nonsymmetric matrices. The gain in speed is achieved by exploiting structure that until now has been ignored. The development of fast methods is important when solving large-scale problems of interest to scientists and engineers. These methods are applicable in the investigation of social networks, whose properties recently have received considerable attention, not only by scientists, but also by the New York Times. A major hurdle in the investigation of social networks is their large size. The research provides improved tools for investigating these kinds of networks. The methods can also be applied in solution methods for Maxwell's equation in three space-dimensions. Essentially, one has to compute the exponential of very large matrices. These matrices are much too large to allow the use of standard software. The methods to be developed within the framework of this proposal speed up the computations by exploiting inherent structure of these problems. Work on the problems of this proposal requires background in linear algebra, orthogonal polynomials and rational functions, approximation theory, and Gauss-type quadrature. Hence the research is well suited for doctoral students working on this project.