Eigenvalues problems, Krylov subspace methods, and subspace recycling

特征值问题、Krylov 子空间方法和子空间回收

• 批准号: 1115520
• 负责人: Daniel Szyld
• 金额: $ 28万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115520&HistoricalAwards=false
• 关键词: Eigenvalues; problems; Krylov; subspace; methods

The project concerns the development, analysis, refinement and testing of efficient numerical algorithms for the solution of algebraic eigenvalue problems and of systems of linear equations arising from a variety of applications. The PI's research is concentrated on the following two classes of problems: 1. Interior eigenvalues of generalized non-Hermitian eigenvalue problems. These arise in many scientific and engineering applications, such as stability analysis of steady flows of incompressible fluid and evaluation of passivity in control systems and circuit networks. 2. Sequences of linear systems of equations. These arise, e.g., in iterative methods for nonlinear problems, such as inexact Newton's method for Riccati equations, inexact eigenvalue algorithms, and interior-point methods for convex optimization. A goal of the project is the development of rapidly convergent and robust Krylov subspace methods to efficiently solve both classes of problems. For the first class of problems, this entails the study of convergence properties, subspace expansion and extraction, and preconditioning techniques that take advantage of the structure of the problems. For the second problem class, the aim is to reduce the iteration counts and computational effort needed for the solution of each linear system by using a properly recycled subspace obtained from the iterative solution of a preceding linear system in the sequence. The study of both problem classes also entails extensive computational experimentation on benchmark problems.The problems to be studied in this project include the efficient computation of a group of eigenvalues and the solution of sequences of linear systems. Eigenvalue calculations include analysis of vibration frequencies in structures including buildings, to make sure, for example, that they are far from the earthquake band. Fast algorithms for generalized eigenvalue problems also contribute to the design and analysis of electronic integrated circuit and micro-electro-mechanical systems (MEMS), and the detection of potential presence of turbulent fluid flows. Efficient solution of a sequence of linear systems facilitates modeling of fatigue and fracture via finite element analysis, and the stability analysis of linear systems through the solution of Riccati equations. The two problems mentioned are fundamental in the field of numerical linear algebra as well as many relevant areas such as fluid and solid mechanics, system and control theory, and numerical optimization. Although numerical algorithms have been developed and studied for some of these problems, efficient solution of large-scale applications remains a major computational challenge. Development and refinement of these computational methods have potential broader impact in engineering and science.

该项目涉及开发、分析、改进和测试有效的数值算法，用于解决代数特征值问题和各种应用产生的线性方程组。PI的研究主要集中在以下两类问题上：1。广义非厄密特征值问题的内特征值。它们出现在许多科学和工程应用中，例如不可压缩流体稳定流动的稳定性分析以及控制系统和电路网络的无源性评估。2. 线性方程组的序列。例如，这些出现在非线性问题的迭代方法中，例如求解Riccati方程的不精确牛顿法，不精确特征值算法和求解凸优化的内点法。该项目的目标是开发快速收敛和鲁棒的Krylov子空间方法来有效地解决这两类问题。对于第一类问题，这需要研究收敛性质，子空间展开和提取，以及利用问题结构的预处理技术。对于第二类问题，目标是通过使用从序列中前一个线性系统的迭代解获得的适当循环子空间来减少每个线性系统解所需的迭代次数和计算量。这两类问题的研究还需要在基准问题上进行广泛的计算实验。本课题研究的问题包括一组特征值的有效计算和线性系统序列的解。特征值计算包括对建筑物等结构的振动频率进行分析，以确保它们远离地震带。广义特征值问题的快速算法也有助于电子集成电路和微机电系统（MEMS）的设计和分析，以及检测潜在存在的湍流流体流动。线性系统序列的有效求解便于通过有限元分析进行疲劳和断裂建模，并通过求解Riccati方程进行线性系统的稳定性分析。这两个问题是数值线性代数领域的基础问题，也是流体与固体力学、系统与控制理论、数值优化等许多相关领域的基础问题。虽然数值算法已经发展和研究了其中一些问题，但大规模应用的有效解决仍然是一个主要的计算挑战。这些计算方法的发展和改进在工程和科学领域具有潜在的更广泛的影响。