Newton Methods for Nonlinear Problems Affine Invariance and Adaptive
This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local
- PDF / 4,246,068 Bytes
- 432 Pages / 439.37 x 666.14 pts Page_size
- 70 Downloads / 192 Views
35
Peter Deuflhard
Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms With 49 Figures
123
Peter Deuflhard Zuse Institute Berlin (ZIB) Takustr. 7 14195 Berlin, Germany and Freie Universität Berlin Dept. of Mathematics and Computer Science deufl[email protected]
Mathematics Subject Classification (2000): 65-01, 65-02, 65F10, 65F20, 65H10, 65H20, 65J15, 65L10, 65L60, 65N30, 65N55, 65P30
ISSN 0179-3632 ISBN 978-3-540-21099-7 (hardcover) e-ISBN 978-3-642-23899-4 ISBN 978-3-642-23898-7 (softcover) DOI 10.1007/978-3-642-23899-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011937965 © Springer-Verlag Berlin Heidelberg 2004, Corrected printing 2006, First softcover printing 2011. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In 1970, my former academic teacher Roland Bulirsch gave an exercise to his students, which indicated the fascinating invariance of the ordinary Newton method under general affine transformation. To my surprise, however, nearly all global Newton algorithms used damping or continuation strategies based on residual norms, which evidently lacked affine invariance. Even worse, nearly all convergence theorems appeared to be phrased in not affine invariant terms, among them the classical Newton-Kantorovich and NewtonMysovskikh theorem. In fact, in those days it was common understanding among numerical analysts that convergence theorems were only expected to give qualitative insight, but not too much of quantitative advice for application, apart from toy problems. This situation left me deeply unsatisfied, from the point of view of both mathematical aesthetics and algorithm design. Indeed, since my first academic steps, my scientific guideline has been and still is that ‘good’ mathematical theory should have a palpable influence on the construction of algorithms, while ‘good’ algorithms should be as firmly as possible backed by a transparently underlying mathematical theory. Only on such a basis, algorithms will be efficient enough to cope with the enormous difficulties of real life problems. In 1972, I started to work along this line
Data Loading...
