Numerical Data Fitting in Dynamical Systems A Practical Introduction
Real life phenomena in engineering, natural, or medical sciences are often described by a mathematical model with the goal to analyze numerically the behaviour of the system. Advantages of mathematical models are their cheap availability, the possibility
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Applied Optimization Volume 77
Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
NUlllerical Data Fitting in Dynalllical Systellls A Practical Introduction with Applications and Software
by
Klaus Schittkowski Department of Mathematics, University of Bayreuth, Bayreuth, Germany
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4757-6050-7 ISBN 978-1-4419-5762-7 (eBook) DOI 10.1007/978-1-4419-5762-7
Printed on acid-free paper
All Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
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Contents
Preface
xi
1. INTRODUCTION
1
2. MATHEMATICAL FOUNDATIONS Optimality Criteria 1
2
3
1.1
7 7 7
1.2 1.3
9 10
Least Squares Methods Optimality Conditions Gauss-Newton and Related Methods Solution of Least Squares Problems by SQP Methods Constrained Least Squares Optimization Alternative Norms Numerical Solution of Ordinary Differential Equations 4.1 Explicit Solution Methods 4.2 Implicit Solution Methods Sensitivity Equations 4.3 4.4 Internal Numerical Differentiation Numerical Solution of Differential Algebraic Equations 5.1 Algebraic Equations 5.2 Index of a Differential Algebraic Equation 5.3 Index Reduction and Drift Effect 5.4 Projection Methods
14 14 16 18 20 23 23 24 27 31 33 38 38 40 43 46 48 48 50 52 55
Notation Convexity and Constraint Qualification Necessary and Sufficient Optimality Criteria Sequential Quadratic Programming Methods 2.1 The Quadratic Programming Subproblem Line Search and Quasi-Newton Updates 2.2 2.3 Convergence 2.4 Systems of Nonlinear Equations
3.1 3.2 3.3 3.4 3.5
4
5
vii
viii
NUMERICAL DATA FITTING IN DYNAMICAL SYSTEMS
5.5 5.6 6
7
8
9
Consistent Initial Values Implicit Solution Methods Numerical Solution of One-Dimensional Partial Differential Equations The General Time-Dependent Model 6.1 Some Special Classes of Equations 6.2 The Method of Lines 6.3 Partial Differential Algebraic Equations 6.4 Difference Formulae 6.5 Polynomial Interpolation 6.6 Upwind Formulae for Hyperbolic Equations 6.7 Essentially Non-Oscillatory Schemes 6.8 Systems of Hyperbolic Equations 6.9 6.10 Sensitivity Equations Laplace Transforms Basic Properties 7.1 7.2 Numerical Back-Transformation Automatic Differentiation 8.1 Forward Mode Reverse Mode 8.2 Statistical Interpretation of Results
3. DATA FITTING MODELS Explicit Model Functions 1 2 Laplace Tra
