Eigenvalues problems, Krylov subspace methods, and subspace recycling

该项目涉及开发、分析、改进和测试用于解决代数特征值问题和由各种应用引起的线性方程组的有效数值算法。PI的研究集中在以下两类问题上：1.广义非埃尔米特特征值问题的内特征值问题。这些问题出现在许多科学和工程应用中，如不可压缩流体定常流动的稳定性分析和控制系统和电路网络中的无源性评估。2.线性方程组序列。例如，这些问题出现在非线性问题的迭代方法中，例如Riccati方程的不精确牛顿方法、不精确特征值算法和凸优化的内点方法。该项目的一个目标是开发快速收敛和健壮的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，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求解每个线性系统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大量的计算实验。本项目要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频带。广义特征值问题的快速算法也有助于电子集成电路和微电子机械系统(MEMS)的设计和分析，以及检测可能存在的湍流流动。一系列线性系统的有效解有助于通过有限元分析对疲劳和断裂进行建模，并通过求解Riccati方程来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以及流体与固体力学、系统与控制理论、数值优化等许多相关领域的基本问题。虽然对其中一些问题已经开发和研究了数值算法，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可能会在工程和科学中产生更广泛的影响。

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.