Generalized Locally Toeplitz Sequences: Theory and Applications

Generalized Locally Toeplitz Sequences: Theory and Applications

Carlo Garoni Stefano Serra-Capizzano Generalized Locally Toeplitz Sequences: Theory and Applications Volume I 123

Carlo Garoni Department of Science and High Technology University of Insubria Como Italy Stefano Serra-Capizzano Department of Science and High Technology University of Insubria Como Italy ISBN 978-3-319-53678-1 ISBN 978-3-319-53679-8 (ebook) DOI 10.1007/978-3-319-53679-8 The present book has been realized with the financial support of the Italian INdAM (Istituto Nazionale di Alta Matematica) and the European Marie-Curie Actions Programme through the Grant PCOFUND-GA-2012-600198. Library of Congress Control Number: 2017932016 Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface Sequences of matrices with increasing size naturally arise in several contexts and especially in the discretization of continuous problems, such as integral and differential equations. The theory of generalized locally Toeplitz (GLT) sequences was developed in order to compute/analyze the asymptotic spectral distribution of these sequences of matrices, which in many cases turn out to be GLT sequences. In this book we present the theory of GLT sequences together with some of its main applications. We will also refer the reader to the available literature for further applications that are not included herein. It normally happens in mathematics that ideas are better conveyed in the univariate setting and then transferred to the multivariate setting by successive generalizations. This is the case with mathematical analysis, for example. Any first course in mathematical analysis focuses on the theory of continuous/differentiable/ integrable functions of one variable, while concepts like multivariate continuous functions, partial derivatives, multiple integrals, etc., are introduced only later (usually in a second course, not in the first one). Something similar occurs here. The present volume is the analog of a first course in mathematical analysis; it addresses the theory of what we may call univariate GLT sequences (or unilevel GLT sequences according to a more traditional terminology). As we shall see, univariate GLT sequences arise in the discretization of unidimensional integral and differential equations. The analog of a second course in mathematical analysis is [62], which deals with multivariate/multilevel GLT sequences, a kind of sequence arising from the discretization of multidimensional integral and differential equations. The generalization to the multivariate setting offered by [62] is certainly fundamental, especially for the applications, but it is essentially a technical matter whose purpose is to implement appropriately the ideas we are already informed of by the present volume. In short, this volume already contains all the ideas of the theory of GLT sequences, just as a first course in mathematical analysis contains all the ideas of integro-differential calculus. v

vi Preface The book is conceptually divided into two parts. The first part (Chaps. 1 8) covers the theory of GLT sequences, which is finally summarized in Chap. 9. The second part (Chap. 10) is devoted to the applications, corroborated by several numerical illustrations. Some exercises are scattered in the text and their solutions are collected in Chap. 12. Each exercise is placed at a particular spot with the idea that the reader at that stage possesses all the elements to solve it. The book is intended for use as a text for graduate or advanced undergraduate courses. It should also be useful as a reference for researchers working in the fields of linear algebra, numerical analysis, and matrix analysis. Given its analytic spirit, it could also be of interest for analysts, primarily those working in the fields of measure and operator theory. The reader is expected to be familiar with basic linear algebra and matrix analysis. Any standard university course on linear algebra covers all that is needed here. Concerning matrix analysis, an adequate preparation is provided by, e.g., [16] or [67]; we refer in particular to [16, Chaps. 1 3, Sects. 1 3 of Chap. 6, and Sects. 1 8 of Chap. 7] and [67, Chap. 2, Sects. 5.5, 7.1 7.2, and 8.1]. In addition, the reader who knows Chaps. 1 4 of Bhatia s book [12] will certainly take advantage of this. Some familiarity with real and complex analysis (especially, measure and integration theory) is also necessary. For our purposes, Rudin s book [95] is more than enough; actually, Chaps. 1 5 of [95] cover almost everything one needs to know. Finally, a basic knowledge of general topology, functional analysis, and Fourier analysis will be of help. Assuming the reader possesses the above prerequisites, most of which will be addressed in Chap. 2, there exists a way of reading this book that allows one to omit essentially all the mathematical details/technicalities without losing the core. This is probably the best way of reading for those who love practice more than theory, but it is also advisable for theorists, who can recover the missing details afterwards. It consists in reading carefully the introduction in Chap. 1 (this is not really necessary but it is recommended), the summary in Chap. 9, and the applications in Chap. 10. To conclude, we wish to express our gratitude to Bruno Iannazzo, Carla Manni, and Hendrik Speleers, who awakened our interest in the theory of GLT sequences and ultimately inspired the writing of this book. We also wish to thank all of our colleagues who worked in the field of Toeplitz matrices and spectral distributions, and contributed with their work to lay the foundations of the theory of GLT sequences. We mention in particular Bernhard Beckermann, Albrecht Böttcher, Fabio Di Benedetto, Marco Donatelli, Leonid Golinskii, Sergei Grudsky, Arno Kuijlaars, Maya Neytcheva, Debora Sesana, Bernd Silbermann, Paolo Tilli, Eugene Tyrtyshnikov, and Nickolai Zamarashkin. Finally, special thanks go to Giovanni Barbarino and Dario Bini, who agreed to read this book and provided useful advice on how to improve the presentation.

Preface vii Based on their research experience, the authors propose a reference textbook in two volumes on the theory of generalized locally Toeplitz sequences and their applications. This first volume focuses on the univariate version of the theory and the related applications in the unidimensional setting, while the second volume, which addresses the multivariate case, is mainly devoted to concrete PDE applications. Como, Italy December 2016 Carlo Garoni Stefano Serra-Capizzano

Contents 1 Introduction... 1 1.1 Main Application of the Theory of GLT Sequences.... 1 1.2 Overview of the Theory of GLT Sequences.... 4 2 Mathematical Background... 7 2.1 Notation and Terminology... 7 2.2 Preliminaries on Measure and Integration Theory... 10 2.2.1 Essential Range.... 10 2.2.2 L p Spaces... 12 2.2.3 Convergence in Measure, a.e., in L p... 14 2.2.4 Riemann-Integrable Functions.... 19 2.3 Preliminaries on General Topology... 20 2.3.1 Pseudometric Spaces... 20 2.3.2 The Topology s measure of Convergence in Measure... 22 2.4 Preliminaries on Matrix Analysis... 27 2.4.1 p-norms.... 27 2.4.2 Singular Value Decomposition... 29 2.4.3 Schatten p-norms... 31 2.4.4 Singular Value and Eigenvalue Inequalities... 34 2.4.5 Tensor Products and Direct Sums... 40 2.4.6 Matrix Functions.... 41 3 Singular Value and Eigenvalue Distribution of a Matrix-Sequence... 45 3.1 The Notion of Singular Value and Eigenvalue Distribution... 45 3.2 Rearrangement... 47 3.3 Clustering and Attraction... 49 3.4 Zero-Distributed Sequences... 52 ix

x Contents 4 Spectral Distribution of Sequences of Perturbed Hermitian Matrices.... 57 4.1 Preliminary Results... 57 4.2 Main Results... 62 5 Approximating Classes of Sequences.... 65 5.1 The a.c.s. Notion... 65 5.2 The a.c.s. Topology s a:c:s:... 66 5.2.1 Construction of s a:c:s:... 67 5.2.2 Expression of d a:c:s: in Terms of Singular Values... 71 5.2.3 Connection Between s a:c:s: and s measure... 72 5.3 The a.c.s. Tools for Computing Singular Value and Eigenvalue Distributions... 74 5.4 The a.c.s. Algebra... 83 5.5 Some Criteria to Identify a.c.s..... 88 5.6 An Extension of the Concept of a.c.s.... 92 6 Toeplitz Sequences... 95 6.1 Toeplitz Matrices and Toeplitz Sequences... 95 6.2 Basic Properties of Toeplitz Matrices.... 97 6.3 Schatten p-norms of Toeplitz Matrices... 100 6.4 Circulant Matrices... 106 6.5 Singular Value and Spectral Distribution of Toeplitz Sequences: An a.c.s.-based Proof... 108 6.6 Extreme Eigenvalues of Hermitian Toeplitz Matrices.... 111 7 Locally Toeplitz Sequences.... 115 7.1 The Notion of LT Sequences... 115 7.2 Properties of the LT Operator... 121 7.3 Fundamental Examples of LT Sequences... 125 7.3.1 Zero-Distributed Sequences.... 125 7.3.2 Sequences of Diagonal Sampling Matrices... 126 7.3.3 Toeplitz Sequences... 130 7.4 Singular Value and Spectral Distribution of a Finite Sum of LT Sequences... 133 7.5 Algebraic Properties of LT Sequences... 135 7.6 Characterizations of LT Sequences.... 136 8 Generalized Locally Toeplitz Sequences... 143 8.1 Equivalent Definitions of GLT Sequences... 143 8.2 Singular Value and Spectral Distribution of GLT Sequences... 144 8.3 Approximation Results for GLT Sequences... 146 8.3.1 Characterizations of GLT Sequences.... 151 8.3.2 Sequences of Diagonal Sampling Matrices... 152

Contents xi 8.4 The GLT Algebra.... 154 8.5 Algebraic-Topological Definitions of GLT Sequences... 163 9 Summary of the Theory... 165 10 Applications.... 173 10.1 The Algebra Generated by Toeplitz Sequences... 173 10.2 Variable-Coefficient Toeplitz Sequences.... 175 10.3 Geometric Means of Matrices... 183 10.4 Discretization of Integral Equations... 185 10.5 Finite Difference Discretization of Differential Equations.... 189 10.5.1 FD Discretization of Diffusion Equations... 191 10.5.2 FD Discretization of Convection-Diffusion-Reaction Equations... 198 10.5.3 FD Discretization of Higher-Order Equations... 210 10.5.4 Non-uniform FD Discretizations... 212 10.6 Finite Element Discretization of Differential Equations... 218 10.6.1 FE Discretization of Convection-Diffusion-Reaction Equations... 218 10.6.2 FE Discretization of a System of Equations... 225 10.7 Isogeometric Analysis Discretization of Differential Equations... 229 10.7.1 B-Spline IgA Collocation Discretization of Convection-Diffusion-Reaction Equations.... 229 10.7.2 Galerkin B-Spline IgA Discretization of Convection-Diffusion-Reaction Equations.... 244 10.7.3 Galerkin B-Spline IgA Discretization of Second-Order Eigenvalue Problems... 254 11 Future Developments... 261 12 Solutions to the Exercises... 265 References... 299 Index... 305