Eigenvalues problems, Krylov subspace methods, and subspace recycling

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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.