Methods in Approximation: Techniques for Mathematical Modelling

238 Pages·1986·5.63 MB·English

by  Richard E. Bellman

Methods in Approximation
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universita degli Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-C. ROTA, M.I. T., Cambridge, Mass., U.S.A.
Methods in
Approximation
Techniques for Mathematical Modelling
Richard E. Bellman
Department ofE lectrical Engineering. University of Southern California.
Los Angeles. U.S.A.;
Center for Applied Mathematics. The University of Georgia.
Athens. Georgia. U.S.A.
and
Robert S. Roth
Boston. U.S.A.
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Libnry or Congress Cataloging in Publication Data
Bcllmun, RichJrd Ernest, 1920-
Methods in Jpproximation.
(Muthcmatic~ and its upplication~)
Includes bibliographies Jnd index.
I. Approximation theory. I. Roth, Robert, 1930-
II. Title. Ill. Series: Mathematics and its upplications
(D. Reidel Publbhing Company)
QA22I.B36 1986 511'.4 86- 461
ISBN·13 :97IJ.94'()1 0·8544·1 e-ISBN-13: 978-94-009-4600-2
001: 10. 1007/978-94-009-4600-2
Published by D. Reidel Publishing Company,
P.O. Box 17, 3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by K1uwer Academic Publishers Group,
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All Rights Reserved
() 1986 by D. Reidel Publishing Company, Dordrecht, HoUand
Softcover reprint of the hardcover 1st edition 1986
No parI of the material protected by this copyright notice may be reproduced or utilized
in any fonn or by any means, electronic or mechanical, including photocopying,
.eeo.dinj: or by any infonnation storage and retrieval system,
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CONTENTS
EDITOR'S PREFACE ix
PREFACE xi
Chapter
1. BASIC CONCEPTS • 1
Introduction • . . .• .•••• 1
Integral Domains,Fields and Vector
Spaces • . • .• .•••• 2
Subspaces, Bases and Inner Products • 4
Spaces, Subspaces and Approximation • 7
The Continuous Function . • . • • . • 8
Polynomial Subspaces • . . • • • . . 9
Spaces Generated by Differential
Equations . . • . • . • • 10
The Piecewise Linear Function • 13
Discussion •• • • . . . 15
Bibliograpy and Comments 16
2. POLYNOMIAL APPROXIMATION • • • . 17
Introduction • . • . • . • • • • • 17
Piecewise Linear Functions • • 22
Curve Fitting by Straight Lines. . 24
A One Dimensional Process in Dynamic
Programming . . . • . • • . • 26
The Functional Equation • . • • •• 27
The Principle of Optimality. • •• 29
A Direct Derivation . . . . • • • • 29
Curve Fitting by Segmented Straight
Lines • . • • . • • • . • 30
A Dynamic Programming Approach 31
A Computational Procedure . 32
Three Dimensional polygonal
Approximation • • • • • • 33
The Orthogonal Polynomials •••• 35
The Approximation Technique 36
CONTENTS
Discussion .•• • 37
Bibliography and Comments 38
3. POLYNOMIAL SPLINES . 40
Introduction . . . . . . . • 40
The Cubic Spline I •..••. 40
Construction of the Cubic Spline 41
Existence and Uniqueness • . • • • 46
A Computational Algorithm - Potter's
Method • . • . . . . • . . . 48
Splines via Dynamic Programming • • 53
Derivation of Splines by Dynamic
Programming • . . • • . • •• 54
Equivalence of the Recursive
Relations obtained by Dynamic
Programming and the Usual
results • . . 59
Cardinal Splines . . . . . 61
Polynomial Splines 63
Generalized Splines . . • . 64
Mean Square Spline Approximation 65
The Cubic Spline II . . . • . 66
The Minimization Procedure 67
The Functional Equation • 68
Recursion Relations . • . . . . . . 69
Bibliography and Comments . . . .. 69
4. QUASILINEARIZATION ••. 71
Introduction •. .. .•... 71
Quasilinearization I ..•. 71
The Newton Raphson Method . 72
Quasilinearization II . • • • • •. 76
Existence . . . • . . • . • 77
Convergence . . . . . . . . • • •. 79
An Example, Parameter Identification 82
Unknown Initial Conditions . . • • 85
Damped Oscillations • • . . • • • . 88
Segmental Differential Approximation 92
Differential Systems with Time
Varying Coefficients . . . • 94
A Method of Solution. . . • • .. 95
An Interesting Case . . • . . • .• 98
Discussion ... . . • • . . 100
Bibliography and Comments . • 101
CONTENTS
5. DIFFERENTIAL APPROXIMATION . 103
Introduction •. . . • . • • 103
Differential Approximation ••.• 103
Linear Differential Operators ••• 104
Degree of Approximation • . • . • • 105
Improving the Approximation . . •• 106
An Example . • . . • • • . . . • • 107
Differential-Difference Equations . 110
A Useful Approximation to g(t) •. 112
Discussion . • . . . • . . • . •• 115
An Example •••...•...•. 117
Functional Differential Equations • 119
The Nonlinear Spring . . . 120
The Van der Pol Equation . • • 122
Bibliography and Comments .•. 124
6. DIFFERENTIAL QUADRATURE . . 125
Introduction •• . . . 125
Differential Quadrature ...... 125
Determination of the Weighting
Coefficient~ ..•..... 126
A First order Problem ....... 129
A Nonlinear Wave Equation ..... 131
Systems of Nonlinear Partial
Differential Equations 136
Higher Order Systems .. ... 139
Long Term Integration ....... 141
G(y) Linear . . . . . . . . . 142
G(y) Nonlinear .........• 142
A Mathematical Problem . . . 143
Systems with Partial Information . 144
Bibliography and Comments ..... 147
7. EXPONENTIAL APPROXIMATION ...... 149
Introduction •• . . . • . •. 149
Approximation in Function Space •• 150
An Example - Pharmacokinetics ... 151
Other Physical Processes . • • 153
Proney's Method .....•..•• 153
The Renewal Equation . . . •• 156
The Fredholm Integral Equation •• 158
Bibliography and Comments ••... 160
viii CONTENTS
8. THE RICCATI EQUATION .••••. 161
Introduction . . • . . . 161
The Linear Differential Equation . 162
Differential Inequalities ...•• 164
Solution of the Riccati Equation in
terms of the Maximum
Operation • . • • • . 166
Upper and Lower Bounds. . • 169
Successive Approximations via
Quasilinearization ....• 170
An Illustrative Example. • •. 170
Higher Order Approximations ••.• 171
Multidimensional Riccati Equation. 173
Variational Problems and the Riccati
Equation ••...•..•. 175
Bibliography and Comments ••... 176
9. SOLUTION OF APPROXIMATE EQUATIONS 178
Introduction •.•...•.... 178
First Order Differential Equations 179
The Second Order Differential
Equation • • . . • .. . 179
Discussion .•• • . • . .• . 182
Linear Perturbations . . •. • 183
The Van der Pol Equation I .... 184
The Van der Pol Equation II •... 186
The Riccati Equation • • . 189
u" + a{t)u = 0 .•.•••.•.• 190
Another Approach • . • . . 193
Discussion •. . . . • . . 197
Bibliography and Comments . . 198
10. MAGNETIC FIELD DETERMINATION. . . 200
Introduction •• . . . • . 200
The Theoretical Problem • . 201
Maxwell's Equations . . • •. . 203
A Variational Principle . • . . . . 206
The Finite Element Method. • 208
Computational Aspects . . • • • • 212
Analytical Considerations 213
Boundary Condi tions • • . . • 216
Discussion •. • • . . . . 220
Bibliography and Comments • 220
. . . . . . . . . . . . .
INDEX • . 222
EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'.
van Gulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the "tree" of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is used (non
trivially) in regional and theoretical economics; algebraic geometry interacts with
physics; the Minkowsky lemma, coding theory and the structure of water meet one
another in packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; Lie algebras are relevant
to filtering; and prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as "experimental
mathematics", "CFD", "completely integrable systems", "chaos, synergetics and
large-scale order", which are almost impossible to fit into the existing classification
schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging
(sub)disciplines and to such (new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical
and/or scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour
into another;
- influences which the results, problems and concepts of one field of enquiry have
and have had on the development of another.
The Mathematics and Its Applications programme tries to make available a careful
selection of books which fit the philosophy outlined above. With such books, which
are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope
to contribute something towards better communication among the practitioners in
diversified fields.
Because of the wealth of scholarly research being undertaken in the Soviet
Union, Eastern Europe, and Japan, it was decided to devote special attention to the
work emanating from these particular regions. Thus it was decided to start three
regional series under the umbrella of the main MIA programme.
ix
x EDITOR'S PREFACE
As the authors stress in their preface there are two kinds of approximations
involved when doing mathematical investigations of real world phenomena. First
there is the idealized mathematical model in its full complexity and then there are
the approximations one must make in order to be able to do something with it.
And, of course, the inaccuracies adhering to the determination of whatever meas
ured constants are involved. Thus,
"In every mathematical investigation the question will arise whether we can
apply our results to the real world.... Consequently, the question arises of
choosing those properties which are not very sensitive to small changes in
the model and thus maybe viewed as properties of the real process.
V.1. Arnol'd, 1978
Such are the considerations which lie at the basis of several fields in mathematics:
deformation theory, perturbation theory and approximation theory. In the setting
above the task of approximation theory becomes that of finding that approximate
model which precisely captures those properties which are not very sensitive to
small changes. This also helps to make it clear that much more is involved than
neglecting small terms or doing an expansion in some ( or throwing away all non
linear terms. Often indeed there will not even be an obvious small quantity.
The senior author of this volume, the late Richard Bellmann, has spent much time
on thinking about the why and how of approximation and has pioneered several
new methods in the field. The book can be accurately described as a survey of his
thoughts on the topic during the last 25 years. Much of the material dates from the
middle and late seventies. Here this material is presented in a coherent fashion by
Bellmann and Roth without losing the typical inventive and stimulating character of
much of Bellmann's writing.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima
gined.
William Blake
Bussum, December 1985 Michiel Hazewinkel