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.
Eigenvalue problems refer to the problem of finding privileged frequencies, and the corresponding structures that are associated with those frequencies. For example, the design of railroad tracks and railroad cars to minimize the noise. For detailed calculations one needs to apply specialized algorithms. In this project, new and existing algorithms were studied. In particular, it was shown under which mathematical circumstances, these methods can converge faster to the desired solutions. Another aspect of this project concerned the study of certain iterative methods for the solution of linear systems. Linear systems are ubiquitous in science and engineering. The most widely used iterative methods require too much computational storage. An alternative method for a particular class of systems (nearly Hermitian) requiring less storage was studied, and shown to be unstable. An alternative stable algorithm was then proposed and analyzed. One way to reduce the storage requirements of these iterative methods is to save information from a solution of a previously solved system, for example, a system with the same coefficient matrix, or one with a coefficient matrix close in some sense to that of the current system. This approach is called "recycling," and has been shown to be efficient in many circumstances. In this project, this idea was explored for sequences of shifted linear systems, which occur, for example in quantum chromodynamics and in certain problems arising in control.