This project investigates high performance algorithms in the areas of model reduction algorithms for dynamical linear systems, eigenproblem solvers, and iterative methods for sparse nonsymmetric linear systems with one or multiple right-hand side vectors. The model reduction work is based on the rational Lanczos algorithm which performs multipoint moment matching-based model reduction with automatic point selection from a specified grid. It uses a novel shifting strategy which exploits a mix of imaginary and real positive scalar values that provide a multiscale view of the frequency response of the dynamical system being modeled. The restriction to moment-matched reduced order models will be relaxed in order to improve the efficiency of the model production. This will entail the development of new error modeling techniques as well as the assessment of the effect of the use of iterative methods to approximately solve the linear systems that occur in each iteration of rational approximation algorithms. This will also drive the development of new preconditioning and iterative method strategies. A strategy for point placement (rather than point selection from a predetermined grid) will be developed and used in order to improve the accuracy of the reduced order model and to drive a load balancing strategy for the large grain parallel processing in a multilevel parallelism implementation. In the work on eigenproblems, technology will be transferred and adapted from the rational Lanczos domain to Krylov-based eigenproblem solvers. This includes adapting the model reduction shift strategy to yield a multilevel approach to locate appropriate sections of the complex plane in which eigenvalues reside. The choice of spaces used to form the projector will also be updated via rational Lanczos technology. A Gershgorin disk-based alternative to Krylov-based approaches will also be studied. The approach is a generalization and improvement of work by Varga and other s. The methods natural multilevel parallelism will be analyzed and exploited in an experimental implementation. The area of nonsymmetric sparse linear system solving via preconditioned iterative methods supports the advances in the two areas above, and also contributes to the state-of-the-art in numerical algorithms. Three basic tasks will be undertaken. The first is to continue work on a robust parallel preconditioned iterative method-based package for the solution of nonsymmetric systems. This work will build on earlier efforts on the EN-like family of methods, partitioned row projection schemes, and an accelerated block Stiefel iteration adapted to nonsymmetric systems. Preconditioners based on eigenvalue deflation, incomplete orthogonalization, and modified Krylov methods will be considered. The second system linear system solving task that will be addressed is the development and analysis of a family of block EN-like methods for linear systems with multiple right-hand side vectors encountered in multiple-input-multiple-output dynamical systems and applications such as electromagnetics. Finally, the linear system solvers above will be adapted to the situation encountered in model reduction -- multiple linear systems defined by a matrix pencil (A,E), and a set of scalar shifts with associated right-hand sides.
