Applications throughout computational science require the solution of large-scale linear systems and eigenvalue problems, tasks that are often accomplished using Krylov subspace projection methods. Nonsymmetric matrices pose a particular challenge, with accurate solutions often requiring a combination of restarted iterations and effective preconditioning. This project seeks to develop an improved understanding of the convergence of restarted Krylov subspace algorithms for linear systems and eigenvalue problems. The behavior of such methods depends not only upon the eigenvalues of the matrices involved, but also on nonnormality and properties of starting vectors. These latter issues complicate analysis and can lead to algorithm failure; new insight into the mechanisms that spawn such failure will inform the design of improved restarted methods. The project will also consider the important role of preconditioners. Projection methods for dimension reduction of large-scale problems raise related concerns. Reduced-order models may capture salient eigenvalues of the original system, yet miss important transient features of the solution that are of physical significance, especially if the model derives from a nonlinear system. Throughout this project test cases will be drawn from applications such as fluid dynamics.
Large-scale linear algebra problems play a central role in many areas of computational science and engineering, with applications ranging from fluid dynamics and circuit simulation to neuroscience and data mining. Though the efficient solution of such problems is essential to high-fidelity mathematical modeling and the nation's fastest computers devote many cycles to this challenge, several of the most important algorithms are unreliable and not yet understood. This project seeks answers to fundamental questions concerning the behavior of such methods, with the goal of gaining insights that will lead to more rapid and reliable algorithms. Given the many fields that rely on these techniques, such improvements will have broad application throughout the scientific computing community. To complement the research program, this project includes an important educational component comprising the mentorship of graduate and undergraduate students, the development of a graduate course, and the broad public dissemination of educational material for numerical analysis, a core discipline for students preparing for careers in computational science and engineering.
