Eigenvalues problems, Krylov subspace methods, and subspace recycling

该项目涉及开发、分析、改进和测试有效的数值算法，以解决代数特征值问题和各种应用中产生的线性方程组。PI的研究集中在以下两类问题上：1。广义非厄米特特征值问题的内特征值。这些问题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中无源性的评估。2.线性方程组序列。这些出现，例如，在非线性问题的迭代方法中，如Riccati方程的不精确牛顿法，不精确特征值算法和凸优化的邻近点方法。 该项目的一个目标是开发快速收敛和鲁棒的Krylov子空间方法，以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，子空间扩展和提取，以及利用问题结构的预处理技术。对于第二类问题，其目的是通过使用从序列中的前一线性系统的迭代解获得的适当回收的子空间来减少每个线性系统的解所需的迭代计数和计算工作量。这两类问题的研究还需要在基准问题上进行大量的计算实验。本项目将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求解。特征值计算包括分析结构（包括建筑物）的振动频率，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，以及湍流流体流动的潜在存在的检测。一系列线性系统的有效解便于通过有限元分析对疲劳和断裂进行建模，以及通过求解Riccati方程对线性系统进行稳定性分析。这两个问题是基本的数值线性代数领域以及许多相关领域，如流体和固体力学，系统和控制理论，数值优化。 虽然数值算法已经开发和研究了这些问题中的一些，大规模应用的有效解决方案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。

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.