Krylov Projection Methodsfor Model

This dissertation focuses on e ciently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The rst algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an e cient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-inputmultiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees. iii DEDICATION To my wife, Kimberly. iv ACKNOWLEDGMENTS I would like to thank all those who supported me throughout my doctoral studies. Foremost on this list are my advisors, Professors Kyle Gallivan and Paul Van Dooren. Paul deserves the credit for starting me on my work in numerical linear algebra. His insights showed me what an interesting eld it can be. Many thanks go to Kyle for his support through the latter years of my doctoral research. His encouragement and humor are much appreciated. I also thank Drs. Danny Sorensen, Steve Ashby and Eli Chiprout for working with me at di erent times over the past three years. Each was an excellent host who contributed signi cantly to my education. Danny played a key role in my introduction to iterative Krylov methods. Steve pointed me in the direction of Davidson's method, an approach that implicitly touches many parts of this dissertation. Eli patiently led me through the basics of circuit analysis and asked many stimulating questions. My gratitude also goes out to my examination committee, Professors Bassam Bamieh, Farid Najm and M. Pai, for their time and comments. Professors Bamieh and Najm also deserve thanks for providing hours of classroom instruction. Finally, I thank the Department of Energy for its nancial support. This dissertation was supported in part by the Computational Science Graduate Fellowship Program of the O ce of Scienti c Computing in the Department of Energy. v TABLE OF CONTENTS CHAPTER PAGE