Accelerating the arnoldi iteration: theory and practice

Accelerating the Arnoldi Iteration { Theory and Practice by Chao Yang The Arnoldi iteration is widely used to compute a few eigenvalues of a large sparse or structured matrix. However, the method may su er from slow convergence when the desired eigenvalues are not dominant or well separated. A systematic approach is taken in this dissertation to address the issue of how to accelerate the convergence of the Arnoldi algorithm within a subspace of limited size. The acceleration strategies presented here are grouped into three categories. They are the method of restarting, the method of spectral transformation and the Newtonlike acceleration. Simply put, the method of restarting repeats a k-step Arnoldi iteration after improving the starting vector. The method is further divided into polynomial and rational restarting based on the way the starting vector is modi ed. We show that both mechanisms can be implemented in an implicit fashion by relating the restarted Arnoldi to a truncated QR or the RQ iteration. The rational restarting via a Truncated RQ (TRQ) iteration converges extremely fast. However, a linear system must be solved for each restarting. The possibility of replacing a direct linear solver with a preconditioned iterative solver while maintaining the rapid convergence of TRQ is explored in this thesis. The method of spectral transformation is based on the idea of transforming the original eigenvalue problem into one that is easier to solve. Again, both polynomial and rational transformations are possible. Practical issues regarding the design and implementation of e ective spectral transformations are discussed. Finally, one can treat the eigenvalue problem as a nonlinear equation upon which Newton-like methods can be applied. The Jacobi-Davidson (JD) algorithm proposed by Sleijpen and Van der Vorst takes this approach. In JD, the Arnoldi iteration is merely used as a global searching tool to provide a good starting point for the Newton iteration. This algorithm shares many similar properties with the TRQ iteration. Numerical comparisons between these two methods are made in this thesis. Acknowledgments I am deeply indebted to my thesis advisor, Professor Dan Sorensen, for his support and friendship. Dan has created many opportunities for me to work on interesting and important problems in scienti c computing. He set a high standard for me and taught me to be resilient in solving di cult research problems. I would also like to thank other members of my thesis committee for their interest in my thesis. I thank Rich Lehoucq for introducing me to the ARPACK software project which led me into this fascinating area of large scale eigenvalue computation. I bene ted from many discussions with him. I am grateful to Ralph Byers for giving me good advice and encouraging me to continue my graduate study at Rice. Special thanks go to John Lewis at the Boeing Company for inviting me to work there on some very interesting aspects of the eigenvalue problems. I would also like to thank Ed Rothberg, Giacomo Brussino and Mimi Celis at Silicon Graphics, Inc. for giving me the opportunity to work on some industrial eigenvalue problems. I wish to thank Kristi Maschho for her help, encouragement and friendship. I would also like to thank Keith Berrier and Denise Huang for reading my thesis thoroughly and pointing out many typos and mistakes. The Department of Computational and Applied Mathematics (CAAM) has provided a stimulating environment and friendly atmosphere during my graduate study. I would like to thank all CAAM students, especially my o cemates Jen Rich and Erica Zimmer, for their friendship and help. Finally, I would like to thank my parents Chuanhou Yang and Yuhua Zhao for their encouragement and support. Many thanks are due to my host family Roy v and Sue Stubbs in Warrensburg, Missouri for their support ever since I came to the States.