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Abstract Numerical Solution of Eigenvalue Problems ABSTRACT Title of dissertation: NUMERICAL SOLUTION OF EIGENVALUE PROBLEMS WITH SPECTRAL TRANSFORMATIONS Fei Xue, Doctor of Philosophy, 2009 Dissertation directed by: Professor Howard C. Elman Department of Computer Science Institute for Advanced Computer Studies This thesis is concerned with inexact eigenvalue algorithms for solving large and sparse algebraic eigenvalue problems with spectral transformations. In many appli- cations, if people are interested in a small number of interior eigenvalues, a spectral transformation is usually employed to map these eigenvalues to dominant ones of the transformed problem so that they can be easily captured. At each step of the eigenvalue algorithm (outer iteration), the matrix-vector product involving the trans- formed linear operator requires the solution of a linear system of equations, which is generally done by preconditioned iterative linear solvers inexactly if the matrices are very large. In this thesis, we study several efficient strategies to reduce the computa- tional cost of preconditioned iterative solution (inner iteration) of the linear systems that arise when inexact Rayleigh quotient iteration, subspace iteration and implic- itly restarted Arnoldi methods are used to solve eigenvalue problems with spectral transformations. We provide new insights into a special type of preconditioner with “tuning” that has been studied in the literature and propose new approaches to use tuning for solving the linear systems in this context. We also investigate other strate- gies specific to eigenvalue algorithms to further reduce the inner iteration counts. Numerical experiments and analysis show that these techniques lead to significant savings in computational cost without affecting the convergence of outer iterations to the desired eigenpairs. NUMERICAL SOLUTION OF EIGENVALUE PROBLEMS WITH SPECTRAL TRANSFORMATIONS by Fei Xue Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2009 Advisory Committee: Professor Howard Elman, Chair/Advisor Professor Radu Balan Professor Elias Balaras Professor David Levermore Professor Dianne O’Leary Professor James Baeder, Dean’s Representative c Copyright by Fei Xue 2009 Acknowledgments I owe great debt to many people who helped me in all aspects of life during the past few years at Maryland. I would like to express sincere gratitude to all of those who made this thesis possible. First and foremost, I am indebted to my advisor, Professor Howard Elman, for providing with me exceptional guidance, encouragement and support throughout my graduate study. He taught me the first course in scientific computing, aroused my interest in this area and gave me an invaluable opportunity to work on an attractive and challenging topic. Our discussion on eigenvalue computation and other branches of scientific computing is always enlightening and rewarding. He was very patient and dedicated to help me improve academic writing, and gave me much freedom to develop my own intuitions, viewpoints and interests. He is not only an extraordinary advisor, but also a nice and lovable friend. I have been deeply inspired by his beliefs and values, both in research and everyday life. I am very grateful to Dr. Melina Freitag and her advisor, Professor Alastair Spence at the University of Bath, with whom I have essentially worked in close collaboration. Their groundbreaking contribution to the study of inexact eigenvalue algorithms is fundamental to my thesis. Many of the ideas I developed in this thesis originated from fruitful discussions with them. Their suggestions and encouragement consider- ably motivated my efforts. I also thank Professor Valeria Simoncini at Universit`adi Bologna for her insightful pioneering work in this area, and her careful reading of the first part of my work. ii I would also like to acknowledge help and support from some professors. Professor Dianne O’Leary taught me three courses in scientific computing and has always been informative and supportive. She made quite some helpful comments on my thesis that help improve the thesis considerably. Professor emeritus G. W. Stewart wrote a book on eigenvalue algorithms which is my mostly used reference. His knowledge and understanding of eigenvalue problems is an encyclopedia to my study of this topic. Professor Daniel Szyld at Temple University, Professor Misha Kilmer at Tufts University and associate professor Eric de Sturler at Virginia Tech also have provided very useful information to my thesis. I owe special thanks to my wife, Dandan Zhao, who has always stood by my side. Her love, encouragement, enthusiasm and optimism gave me great happiness, strength and confidence through my pursuit of career. Almost everything I achieved today is due to her everlasting support in our daily life. Words can never express my gratitude to her. Thanks to my parents, who cultivated my interest and dedication in research. Though they are thousands of miles away, their unlimited love and care have been great motivation for my career endeavors. Finally, I would also like to thank some people at the University of Maryland. Thanks to Yi Li for being my best roomate for three years. Two good friends of mine, Ning Jiang and Weigang Zhong, who graduated from the AMSC program a few years ago, were very helpful during my first two years at Maryland. The AMSC program coordinator Alverda McCoy, business manager Sharon Welton and payroll coordinator Jodie Gray gave me significant help in administrative issues through my graduate study. iii Table of Contents List of Tables vii List of Figures viii 1 Introduction 1 2 Background 10 2.1 Basic definitions and tools of eigenvalue problems . ...... 10 2.2 The framework of Krylov subspace projection methods . ..... 13 2.2.1 Definition and basic properties of Krylov subspaces . .... 13 2.2.2 The Arnoldi and Lanczos processes . 14 2.2.3 Someprojectionmethods. 15 2.3 Eigenvaluealgorithms .......................... 17 2.4 Spectraltransformations . 20 2.5 Preconditioned Krylov subspace linear solvers . ...... 22 2.6 Relatedwork ............................... 26 2.7 Preconditioning with tuning . 29 3 Inexact Rayleigh quotient iteration 31 3.1 Introduction................................ 31 3.2 Preliminaries ............................... 34 3.3 Convergence of MINRES in inexact RQI . 37 3.3.1 Unpreconditioned MINRES . 37 3.3.2 Preconditioned MINRES with no tuning . 44 3.3.3 Preconditioned MINRES with tuning . 47 3.3.4 Comparison of SYMMLQ and MINRES used in RQI . 50 3.4 Preconditioner with tuning based on a rank-2 modification...... 52 3.5 NumericalExperiments. 53 3.5.1 Stopping criteria for inner iterations . .. 54 3.5.2 Resultsandcomments . 55 iv 3.6 Sometechnicaldetails .......................... 59 3.6.1 Proof of initial slow convergence . 59 3.6.1.1 Unpreconditioned MINRES . 59 3.6.1.2 Preconditioned MINRES with tuning . 64 3.6.2 AssumptionofTheorem3.3.2 . 65 3.7 Concludingremarks............................ 66 4 Inexact subspace iteration 68 4.1 Introduction................................ 68 4.2 Inexact Subspace Iteration and Preliminary Results . ....... 71 4.2.1 InexactSubspaceIteration . 71 4.2.2 Blockeigen-decomposition . 72 4.2.3 Toolstomeasuretheerror . 74 4.3 Convergence Analysis of Inexact Subspace Iteration . ....... 77 4.3.1 Unpreconditioned and preconditioned block-GMRES with no tuning ............................... 78 4.3.2 Preconditioned block-GMRES with tuning . 81 4.3.3 A general strategy for the phase I computation . 86 4.4 Additional strategies to reduce inner iteration cost . ......... 88 4.4.1 Deflation of converged Schur vectors . 88 4.4.2 Special starting vector for the correction equation . ...... 90 4.4.3 Linear solvers with recycled subspaces . 93 4.5 Numericalexperiments . .. .. 94 4.6 Concludingremarks. .. .. 103 5 Inexact implicitly restarted Arnoldi method 104 5.1 Introduction................................ 104 5.2 Review: the implicitly restarted Arnoldi method . ...... 107 5.3 New strategies for solving linear systems in inexact IRA ........ 112 5.3.1 Thenewtuningstrategy . 113 5.3.2 A two-phase strategy to solve the linear systems in Arnoldi steps117 5.3.3 Linear solvers with subspace recycling for the correction equation120 5.4 A refined analysis of allowable errors in Arnoldi steps . ....... 121 5.5 NumericalExperiments. 128 5.6 Concludingremarks. .. .. 139 v 6 Conclusions and future work 141 Bibliography 144 vi List of Tables 3.1 Comparison of three MINRES methods in the third outer iteration onProblem1 ............................... 57 3.2 Comparison of three MINRES methods in the third outer iteration onProblem2 ............................... 57 3.3 Numbers of preconditioned MINRES iteration steps without tuning (m) needed to have θ1 < 0 ......................... 60 3.4 Number of preconditioned MINRES iteration steps needed to satisfy the stopping criterion in the third outer iteration for Problem2 ... 60 3.5 Number of preconditioned MINRES iteration steps needed to satisfy the stopping criterion in the third outer iteration for Problem3 ... 60 4.1 Parameters used to solve the test problems . .. 96 4.2 Number of preconditioned matrix-vector products for different solution strategyforeachproblem . 101 5.1 Parameters used to solve the test problems . 131 5.2 Inner iteration counts for different solution strategy for each problem 138 vii List of Figures 3.1 MINRES linear residual, MINRES and SYMMLQ eigenvalue resid- ual in the third
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