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Solving Regularized Total Least Squares Problems Based on Eigenproblems Solving Regularized Total Least Squares Problems Based on Eigenproblems Vom Promotionsausschuss der Technischen Universität Hamburg-Harburg zur Erlangung des akademischen Grades Doktor der Naturwissenschaften genehmigte Dissertation von Jörg Lampe aus Hamburg 2010 Lampe, Jörg: Solving Regularized Total Least Squares Problems Based on Eigenproblems / Jörg Lampe. – Als Ms. gedr.. – Berlin : dissertation.de – Verlag im Internet GmbH, 2010 Zugl.: Hamburg-Harburg, Techn. Univ., Diss., 2010 ISBN 978-3-86624-504-4 1. Gutachter: Prof. Dr. Heinrich Voß 2. Gutachter: Prof. Dr. Zdeněk Strakoš 3. Gutachter: Prof. Dr. Lothar Reichel Tag der mündlichen Prüfung: 13.05.2010 Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. dissertation.de – Verlag im Internet GmbH 2010 Alle Rechte, auch das des auszugsweisen Nachdruckes, der auszugsweisen oder vollständigen Wiedergabe, der Speicherung in Datenverarbeitungs- anlagen, auf Datenträgern oder im Internet und der Übersetzung, vorbehalten. Es wird ausschließlich chlorfrei gebleichtes Papier (TCF) nach DIN-ISO 9706 verwendet. Printed in Germany. dissertation.de - Verlag im Internet GmbH URL: http://www.dissertation.de Ich möchte mich bei den Mitarbeitern des Instituts für Numerische Simulation an der TUHH für die schönen letzten Jahre bedanken. In zahlreichen Diskussionen, Vorträgen, Übungsgruppen und Vorlesungen habe ich jede Menge dazugelernt, in erster Linie natürlich fachlich, und manches Mal auch nicht ganz so fachlich. Die Arbeit am Institut sowie die Fertigstellung dieses Buches haben mir stets sehr viel Freude bereitet. Ein besonderer Dank gilt Prof. Heinrich Voß, der mich stets unterstützt hat und immer ein offenes Ohr für mich hatte. Das größte Dankeschön geht an meine Frau Katrin, die mir besonders in dieser Zeit seelisch eine starke Stütze war – sowie an unseren Sohnemann Nils, der mir auch an stressigen Tagen ein Lächeln auf das Gesicht zaubern kann. Contents 1 Introduction 1 2 Regularized Least Squares Problems 3 2.1LeastSquaresProblems......................... 4 2.2Ill-conditionedLSProblems....................... 6 2.2.1 Rank-DeficientProblems..................... 6 2.2.2 DiscreteIll-PosedProblems................... 6 2.3RLSwithoutRegularizationMatrix................... 9 2.3.1 TruncatedSVD.......................... 9 2.3.2 IterativeMethods......................... 10 2.4RegularizationMatrixL ......................... 12 2.5RLSwithRegularizationMatrix..................... 21 2.5.1 TruncatedGeneralizedSVD................... 21 2.5.2 TikhonovRegularizationforLSProblems........... 23 2.5.3 HybridMethods.......................... 25 2.6QuadraticallyConstrainedLS...................... 26 2.6.1 Trust-RegionProblems...................... 27 2.6.2 Solution by one Quadratic Eigenvalue Problem . 32 2.6.3 LSTRSMethod.......................... 35 2.7DetermineHyperparameterforLS.................... 49 2.7.1 DiscrepancyPrinciple....................... 51 2.7.2 InformationCriteria....................... 52 2.7.3 GeneralizedCrossValidation................... 52 2.7.4 L-curve.............................. 52 2.8Summary................................. 57 3 Regularized Total Least Squares Problems 59 3.1TotalLeastSquares............................ 60 3.1.1 TLSProblem........................... 60 3.1.2 CoreProblem........................... 63 3.1.3 ExtensionsofTLS........................ 67 3.2RTLSwithoutRegularizationMatrix.................. 68 3.2.1 TruncatedTotalLeastSquares.................. 68 3.2.2 KrylovMethodsforTLS..................... 69 i Contents 3.3RTLSwithRegularizationMatrix.................... 70 3.3.1 TikhonovRegularizationforTLS................ 71 3.3.2 AboutHybridMethodsinRTLS................ 75 3.3.3 DualRTLS............................ 76 3.4QuadraticallyConstrainedTLS..................... 77 3.4.1 RTLSFixedPointIteration................... 80 3.4.2 RTLSQEP............................. 89 3.4.3 RTLSEVP............................. 115 3.5DetermineHyperparameterforTLS................... 136 3.5.1 L-curveforRTLS......................... 137 3.5.2 SequenceofRTLSProblems................... 138 3.5.3 NumericalExamples....................... 139 3.6Summary................................. 145 4 Conclusions and Outlook 147 A Appendix 149 ii List of Figures 2.1 Contourplot of Ψ(x) with H ≥ 0 and g ⊥S1 .............. 29 2.2 Contourplot of Ψ(x) with indefinite H and g ⊥S1 ........... 29 2.3Secularequation............................. 38 ∗ 3.1 Contourplot of f(x) – hard case with ∆ > Lxmin .......... 87 ∗ 3.2 Contourplot of f(x) – potential hard case g ⊥S1 ........... 87 3.3 Behavior of QEP(f(x)) and W (f(x)) –hardcase........... 94 3.4 Behavior of QEP(f(x)) and W (f(x)) – potential hard case . 94 3.5OccurrenceofnegativeLagrangemultipliers.............. 100 3.6 Convergence history solving QEPs by Li&Ye algorithm . 110 3.7ConvergencehistorysolvingQEPsbySOAR.............. 110 3.8 Convergence history solving QEPs by Nonlinear Arnoldi (exact) . 111 3.9 Convergence history solving QEPs by Nonlinear Arnoldi (early) . 111 3.10 Convergence history solving QEPs by Nonlinear Arnoldi (very early) 112 3.11 Jump below zero of g(θ) inthehardcase................ 125 3.12 Continuous function g˜(θ) inthehardcase............... 126 3.13 Plot of a typical function g(θ) ...................... 131 3.14ConvergencehistorysolvingEVPsbyNonlinearArnoldi....... 134 3.15ConvergencehistoryofRTLSQEPL-curve............... 144 iii List of Figures iv List of Algorithms 2.1LSTRSmethod.............................. 40 2.2Adjustmentstep.............................. 41 2.3NonlinearArnoldiforEVPs....................... 42 2.4NonlinearArnoldiforQEPs....................... 56 3.1RTLSwithquadraticequalityconstraint................ 81 3.2RTLSwithquadraticinequalityconstraint............... 84 3.3RTLSQEP................................. 90 3.4NonlinearArnoldi............................. 106 3.5RTLSEVP................................. 117 4.1InverseIteration.............................. 151 4.2ResidualInverseIteration........................ 152 v List of Algorithms vi Chapter 1 Introduction Inverse problems often arise in engineering praxis. They originate from various fields like acoustics, optics, computerized tomography, image restoration, signal processing or statistics. These problems typically lead to overdetermined linear systems Ax ≈ b, A ∈ Rm×n,b∈ Rm,m≥ n. In the classical least squares approach the system matrix A is assumed to be free from error, and all errors are confined to the observation vector b. However, in engineering applications this assumption is often unrealistic. For example, if the matrix A is only available by measurements or if A is an idealized approximation of the underlying continuous operator then both the matrix A and the right-hand side b are contaminated by some noise. An appropriate approach to this problem is the total least squares (TLS) method which determines perturbations ∆A to the coefficient matrix and ∆b to the vector b such that it holds 2 [∆A, ∆b] F =min! subject to (A +∆A)x = b +∆b, where ·F denotes the Frobenius norm of a matrix. The name total least squares appeared only recently in the literature, i.e. in [33] from Golub and Van Loan in 1980. But under the names orthogonal regression or errors-in-variables this fitting method has a long history in the statistical literature. The univariate case n =1was already discussed in 1877 by Adcock [1]. In 1991 Van Huffel and Vandewalle wrote a fundamental book about total least squares [122]. In this thesis our focus is on ill-conditioned problems which arise for example from the discretization of ill-posed problems such as integral equations of the first kind. Then least squares or total least squares methods often yield physically meaningless solutions, and regularization is necessary to stabilize the computed solution. The approach that is examined more closely in this work is to add a quadratic constraint to the TLS problem. This yields the regularized total least squares (RTLS) problem 2 ≤ [∆A, ∆b] F =min! subject to (A +∆A)x = b +∆b, Lx ∆, 1 Chapter 1 Introduction where ∆ > 0 is a regularization parameter and the regularization matrix L ∈ Rp×n, p ≤ n defines a (semi-)norm on the solution through which the size of the solution is bounded or a certain degree of smoothness can be imposed on the solution. The thesis consists of two parts. In chapter 2 the errors are assumed in the right-hand side b only, and in chapter 3 total least squares and regularized total least squares problems are discussed. Chapter 2 starts with a brief introduction of least squares problems. Then we inves- tigate different types of regularization matrices since they play an important role in most of the regularization methods. Quadratically constrained least squares prob- lems are discussed in more detail which will serve as a suitable background for chapter 3. In subsection 2.6.3 the LSTRS method for solving trust-region subproblems (and thus regularized least squares problems as well) is accelerated substantially by the heavy reuse of information. Further we present several methods for the calculation of a reasonable hyperparameter with special emphasis on the L-curve. Chapter 3 begins by introducing the total least squares problem. We review basic theory and give an overview of several extensions. Since the focus is on discrete ill-posed problems some regularization
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