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Robust Solution Methods for Nonlinear Eigenvalue Problems Robust solution methods for nonlinear eigenvalue problems Th`ese pr´esent´ee le 29 aoutˆ 2013 a` la Facult´e des Sciences de Base Chaire d’algorithmes num´eriques et calcul haute performance Programme doctoral en Math´ematiques Ecole´ polytechnique f´ed´erale de Lausanne pour l’obtention du grade de Docteur `es Sciences par Cedric Effenberger Jury: Prof. Philippe Michel, pr´esident du jury Prof. Daniel Kressner, directeur de th`ese Prof. Peter Benner, rapporteur Prof. Wolf-Jurgen¨ Beyn, rapporteur Prof. Marco Picasso, rapporteur Lausanne, Suisse 2013 Acknowledgments iii Acknowledgments The research documented within this thesis was conducted from November 2009 until February 2012 at the Seminar for Applied Mathematics (SAM) of ETH Zurich and from March 2012 until September 2013 at the Mathematics Institute of Com- putational Science and Engineering (MATHICSE) of EPF Lausanne. Special thanks are due to my thesis advisor, Prof. Dr. Daniel Kressner, for his enthusiasm and support over the past years. I learned a lot from him, both scientif- ically and non-scientifically, and it was him who sparked my interest in nonlinear eigenvalue problems. Furthermore, I thank Prof. Dr. Peter Benner, Prof. Dr. Wolf-Jurgen¨ Beyn, Prof. Dr. Philippe Michel, and Prof. Dr. Marco Picasso for agreeing to serve on my PhD committee. I would like to express my gratitude to Prof. Dr. Heinrich Voß for his interest in my work, for several fruitful discussions about Jacobi-Davidson algorithms, and the invitation to visit the Hamburg University of Technology in December 2012. I am also indebted to Prof. Dr. Karl Meerbergen and Prof. Dr. Wim Michiels for our collaboration and the invitation to visit the KU Leuven in March 2013. Likewise, I am grateful to Prof. Dr. Olaf Steinbach and Dr. Gerhard Unger for our joint work on interpolation-based methods for boundary-element discretizations of PDE eigenvalue problems. Finally, I thank Michael Steinlechner for providing a code basis for the numeri- cal experiments in Chapter 5 as part of his Bachelor project. Financial support by the Swiss National Science Foundation under the SNSF research module Robust numerical methods for solving nonlinear eigenvalue prob- lems within the SNSF ProDoc Efficient Numerical Methods for Partial Differential Equations is thankfully acknowledged. At the private level, I wish to thank all of my colleagues at SAM and MATHICSE for making my stay at these institutes a great experience. In particular, I appre- ciated the always pleasant atmosphere created by my office mates, Andrea, Chris- tine, Elke, Jingzhi, Marie, Petar, Shipeng, and Sohrab. Besides, special thanks go to Christine for many interesting conversations about virtually everything, to Roman for his subtle humor and always getting straight to the point, and to Meiyue for sharing a different brain teaser every day. Last but not least, I am grateful to my parents and my sister for always believing in me, and to Carolin for her support and for scrutinizing the manuscript. Abstract v Abstract In this thesis, we consider matrix eigenvalue problems where the eigenvalue pa- rameter enters the problem in a nonlinear fashion but the eigenvector continues to enter linearly. In particular, we focus on the case where the dependence on the eigenvalue is non-polynomial. Such problems arise in a variety of applications, some of which are portrayed in this work. Thanks to the linearity in the eigenvectors, the type of nonlinear eigenvalue problems under consideration exhibits a spectral structure which closely resembles that of a linear eigenvalue problem. Nevertheless, there also exist fundamental differences; for instance, a nonlinear eigenvalue problem can have infinitely many eigenvalues, and its eigenvectors may be linearly dependent. These issues render nonlinear eigenvalue problems much harder to solve than linear ones, and even though corresponding algorithms exist, their performance comes nowhere close to that of a modern linear eigensolver. Recently, minimal invariant pairs have been proposed as a numerically robust means of representing a portion of a nonlinear eigenvalue problem’s spectral struc- ture. We compile the existing theory on this topic and develop this concept further. Other major theoretical contributions include a deflation technique for nonlinear eigenvalue problems, a first-order perturbation analysis for simple invariant pairs, as well as the characterization of the generic bifurcations of a real nonlinear eigen- value problem depending on one real parameter. Based on these advances in theory, we propose and analyze several new algo- rithms for the solution of nonlinear eigenvalue problems, which improve on the properties of the existing solvers. Various algorithmic details, such as the efficient solution of the occurring linear systems or the choice of certain parameters, are discussed for the proposed methods. Finally, we apply prototype implementations to a range of selected benchmark problems and comment on the results. Keywords: nonlinear eigenvalue problem, minimal invariant pair, numerical con- tinuation, polynomial interpolation, deflation, preconditioned eigensolver, Jacobi- Davidson method Zusammenfassung vii Zusammenfassung Diese Dissertation behandelt Matrixeigenwertprobleme, bei denen der Eigenwert- parameter nichtlinear, der Eigenvektor jedoch weiterhin nur linear auftritt. Insbe- sondere konzentrieren wir uns auf den Fall einer nicht-polynomiellen Abhangigkeit¨ vom Eigenwert. Solche Probleme treten in einer Vielzahl von Anwendungen auf, von denen einige in der Arbeit behandelt werden. Infolge der linearen Abhangigkeit¨ vom Eigenvektor, ist die spektrale Struktur dieser Art von nichtlinearen Eigenwertproblemen der von linearen Eigenwertpro- blemen sehr ahnlich.¨ Allerdings gibt es auch fundamentale Unterschiede. So kann ein nichtlineares Eigenwertproblem zum Beispiel unendlich viele Eigenwerte besit- zen oder seine Eigenvektoren konnen¨ linear abhangig¨ sein. Diese Schwierigkeiten verkomplizieren die Losung¨ nichtlinearer Eigenwertprobleme gegenuber¨ linearen deutlich. Zwar existieren entsprechende Algorithmen, deren Leistungsfahigkeit¨ er- reicht jedoch bei weitem nicht die eines modernen linearen Eigenwertlosers.¨ Die kurzlich¨ vorgeschlagenen minimalen invarianten Paare erlauben eine nu- merisch stabile Darstellung eines Teils der spektralen Struktur nichtlinearer Eigen- wertprobleme. Wir stellen die vorhandene Theorie zu diesem Thema zusammen und entwickeln das Konzept weiter. Ein Deflationsverfahren fur¨ nichtlineare Eigen- wertprobleme, eine Storungsanalyse¨ erster Ordnung fur¨ einfache invariante Paare sowie die Charakterisierung der generischen Bifurkationen eines reellen nichtlinea- ren Eigenwertproblems mit einem reellen Parameter stellen weitere bedeutende Beitrage¨ zur Theorie dar. Gestutzt¨ auf diese theoretischen Fortschritte, entwickeln und analysieren wir mehrere neue Algorithmen zur Losung¨ nichtlinearer Eigenwertprobleme, welche die Eigenschaften der bestehenden Loser¨ verbessern. Auf algorithmische Aspekte, wie die effiziente Losung¨ der auftretenden linearen Systeme oder die Wahl gewis- ser Parameter, gehen wir dabei detailliert ein. Abschließend erproben wir proto- typische Implementationen unserer Verfahren anhand ausgewahlter¨ Testprobleme und kommentieren die Ergebnisse. Schlagworter:¨ nichtlineares Eigenwertproblem, minimales invariantes Paar, nu- merische Fortsetzung, Polynominterpolation, Deflation, vorkonditionierter Eigen- wertloser,¨ Jacobi-Davidson Verfahren Contents ix Contents 1 Introduction 1 1.1 Linear vs. nonlinear eigenvalue problems................2 1.2 Existing solution algorithms.......................4 1.2.1 Newton-based methods......................4 1.2.2 Methods based on contour integration..............5 1.2.3 Infinite Arnoldi methods.....................6 1.2.4 Interpolation-based methods...................6 1.2.5 Methods based on Rayleigh functionals.............7 1.3 Contributions of this thesis........................8 2 Applications of nonlinear eigenvalue problems 11 2.1 Stability analysis of time-delay systems................. 11 2.2 Photonic crystals............................. 12 2.3 Boundary-element discretizations.................... 14 2.4 Vibrating string with elastically attached mass............. 16 3 Minimal invariant pairs 19 3.1 Basic principles.............................. 19 3.1.1 Bounds on the minimality index................. 21 3.1.2 Connection to the eigensystem.................. 22 3.2 Characterizing minimality using non-monomial bases......... 25 3.3 Invariance and minimality of composite pairs.............. 27 3.3.1 A generalization of divided differences.............. 28 3.3.2 Composite pairs with coupling.................. 30 3.3.3 Derivatives of block residuals................... 31 3.4 Extraction and embedding of minimal invariant pairs......... 34 3.4.1 An application........................... 36 x Contents 4 Continuation of minimal invariant pairs 41 4.1 Characterization of minimal invariant pairs............... 42 4.1.1 Characterization of simple invariant pairs............ 43 4.1.2 Characterization of non-simple invariant pairs.......... 43 4.1.3 Characterization of generic bifurcations............. 45 4.2 A pseudo-arclength continuation algorithm............... 53 4.2.1 Pseudo-arclength continuation.................. 53 4.2.2 Predictor.............................. 54 4.2.3 Corrector............................. 55 4.2.4 Solving the linear systems.................... 56 4.2.5 Step size control.........................
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