CAREER: Design and Analysis of Restarted Iterative Methods for Linear Systems, Eigenvalue Problems, and Model Reduction

计算科学中的应用需要求解大规模线性系统和特征值问题，这些任务通常使用Krylov子空间投影方法来完成。非对称矩阵提出了一个特别的挑战，精确的解通常需要重新开始的迭代和有效的预处理的组合。 这个项目旨在提高对线性系统和特征值问题的重新启动Krylov子空间算法的收敛性的理解。 这种方法的行为不仅取决于所涉及的矩阵的特征值，而且还取决于非正规性和性质的起始向量。 后面这些问题使分析变得复杂，并可能导致算法失败;对产生此类失败的机制的新见解将为改进的重新启动方法的设计提供信息。 该项目还将考虑预处理器的重要作用。 投影方法降维的大规模问题引起了相关的关注。 降阶模型可能捕获原始系统的显着特征值，但错过了具有物理意义的解的重要瞬态特征，特别是如果模型来自非线性系统。 在整个项目中，测试案例将从流体动力学等应用中抽取。大规模线性代数问题在计算科学和工程的许多领域中发挥着核心作用，其应用范围从流体动力学和电路模拟到神经科学和数据挖掘。 虽然这些问题的有效解决方案对于高保真数学建模至关重要，并且全国最快的计算机投入了许多周期来应对这一挑战，但一些最重要的算法是不可靠的，并且尚未被理解。 该项目寻求有关此类方法行为的基本问题的答案，目的是获得将导致更快速和可靠算法的见解。 鉴于许多领域都依赖于这些技术，这些改进将在整个科学计算界得到广泛应用。 为了补充研究计划，该项目包括一个重要的教育组成部分，包括研究生和本科生的指导，研究生课程的开发，以及数值分析教育材料的广泛公开传播，这是学生准备从事计算科学和工程职业的核心学科。

Applications throughout computational science require the solutionof large-scale linear systems and eigenvalue problems, tasks thatare often accomplished using Krylov subspace projection methods.Nonsymmetric matrices pose a particular challenge, with accurate solutions often requiring a combination of restarted iterations and effective preconditioning. This project seeks to develop an improved understanding of the convergence of restarted Krylov subspace algorithms for linear systems and eigenvalue problems. The behavior of such methods depends not only upon the eigenvalues of the matrices involved, but also on nonnormality and properties of starting vectors. These latter issues complicate analysis and can lead to algorithm failure; new insight into the mechanisms that spawn such failure will inform the design of improved restarted methods. The project will also consider the important role of preconditioners. Projection methods for dimension reduction of large-scale problems raise related concerns. Reduced-order models may capture salient eigenvalues of the original system, yet miss important transient features of the solution that are of physical significance, especially if the model derives from a nonlinear system. Throughout this project test cases will be drawn from applicationssuch as fluid dynamics.Large-scale linear algebra problems play a central role in many areasof computational science and engineering, with applications ranging from fluid dynamics and circuit simulation to neuroscience and data mining. Though the efficient solution of such problems is essential to high-fidelity mathematical modeling and the nation's fastest computers devote many cycles to this challenge, several of the most important algorithms are unreliable and not yet understood. This project seeks answers to fundamental questions concerning the behavior of such methods, with the goal of gaining insights that will lead to more rapid and reliable algorithms. Given the many fields that rely on these techniques, such improvements will have broad application throughout the scientific computing community. To complement the research program, this project includes an important educational component comprising the mentorship of graduate and undergraduate students, the development of a graduate course, and the broad public dissemination of educational material for numerical analysis, a core discipline for students preparing for careers in computational science and engineering.