The Mollification Method and the Numerical Solution of Ill-Posed Problems

s AND THE NUMERICAL SOLUTION OF ILL-POSED PROBLEMS

THE MOLLIFICATION METHOD AND THE NUMERICAL SOLUTION OF ILL-POSED PROBLEMS

D I E G O A. M U R I O D e p a r t m e n t of Mathematical Sciences University of Cincinnati

A Wiley-Interscience Publication J O H N WILEY & SONS, INC. New York

.

Chichester

.

Brisbane

.

Toronto

.

Singapore

This text is printed on acid-free paper. Copyright © 1993 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012. Library of Congress Cataloging in Publication Data: Murio, Diego Α., 1944The mollification method and the numerical solution of ill-posed problems / by Diego A. Murio. p. cm. "A Wiley interscience publication." Includes bibliographical references and index. ISBN 0-471-59408-3 1. Inverse problems (Differential equations)—Improperly posed problems. 2. Inverse problems (Differential equations)—Numerical solutions. I. Title. QA377.M947 1993 515'.353—dc20 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

93-163

To the musicians in my life: Diego S., Veronica, and Francis.

CONTENTS

Preface Acknowledgments 1. Numerical Differentiation 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8.

Description of the Problem, 1 Stabilized Problem, 4 Differentiation as an Inverse Problem, 9 P a r a m e t e r Selection, 11 Numerical Procedure, 12 Numerical Results, 13 Exercises, 17 References and Comments, 19

2. Abel's Integral Equation 2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Description of the Problem, 22 Stabilized Problems, 27 Numerical Implementations, 38 Numerical Results and Comparisons, 49 Exercises, 55 References and Comments, 57

3. Inverse Heat Conduction Problem 3.1. 3.2.

One-Dimensional I H C P in a Semi-infinite Body, 60 Stabilized Problems, 62

viii

CONTENTS

3.3. 3.4. 3.5. 3.6. 3.7. 3.8.

One-Dimensional I H C P with Finite Slab Symmetry, 72 Finite-Difference Approximations, 77 Integral Equation Approximations, 85 Numerical Results, 88 Exercises, 101 References and Comments, 103

4. Two-Dimensional Inverse Heat Conduction Problem 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

Two-Dimensional I H C P in a Semi-infinite Slab, 107 Stabilized Problem, 109 Numerical Procedure and Error Analysis, 113 Numerical Results, 117 Exercises, 129 References and Comments, 130

5. Applications of the Space M a r c h i n g Solution of the I H C P 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12.

131

Identification of Boundary Source Functions, 131 Numerical Procedure, 135 I H C P with Phase Changes, 141 Description of the Problems, 143 Numerical Procedure, 146 Identification of the Initial T e m p e r a t u r e Distribution, 155 Semi-infinite Body, 155 Finite Slab Symmetry, 157 Stabilized Problems, 159 Numerical Results, 161 Exercises, 165 References and Comments, 166

6. Applications of Stable Numerical Differentiation Procedures 6.1. 6.2. 6.3. 6.4.

107

Numerical Identification of Forcing Terms, 169 Stabilized Problem, 170 Numerical Results, 173 Identification of the Transmissivity Coefficient in the One-Dimensional Elliptic Equation, 175 6.5. Stability Analysis, 177 6.6. Numerical Method, 180 6.7. Numerical Results, 185 6.8. Identification of the Transmissivity Coefficient in the One-Dimensional Parabolic Equation, 189 6.9. Stability Analysis, 190 6.10. Numerical Method, 194 6.11. Numerical Results, 199

169

CONTENTS

ix

6.12. Exercises, 203 6.13. References and Comments, 205 Appendix A Mathematical Background A.l. A.2. A.3. A.4. A.5. A.6. A.7.

208

L" Spaces, 208 T h e Hubert Space L (D.\ 211 Approximation of Functions in Ζ, (Ω), 214 Mollifiers, 217 Fourier Transform, 220 Discrete Functions, 224 References and Comments, 230 2

2

Appendix B. References to the Literature on the I H C P

232

Index

249

PREFACE

During the last 20 years, the subject of ill-posed problems has expanded from a collection of individual techniques to a highly developed and rich branch of applied mathematics. This textbook essentially builds, from basic mathematical concepts, the understanding of the most important aspects of the numerical treatment of the applied inverse theory. T h e subject has g r o w n — a n d continues to grow—at such a fast pace that it is impossible to offer a complete treatment in an introductory textbook and all that can b e d o n e is to discuss a few important and interesting topics. Inevitably, in making the selection, I have been influenced by my own interests which, on the other hand, allowed me the pleasure to write about the particular problems with which I am most familiar. This book is intended to be a self-contained presentation of practical computational methods which have b e e n extensively and successfully applied to a wide range of ill-posed problems. T h e nature of the subject demands the application of special mathematical techniques—rarely seen in typical science courses and strange to normal engineering curricula—with which it is initially difficult to relate the steps of a calculation with the more classical concepts of stability and accuracy. This book is intended to solve the problem by giving an account of the theory that builds from the p h e n o m e n a to be explained, keeping everything in as elementary a level as possible, making it useful to a wide circle of readers. T h e primary goal of this book is to provide an introduction to a n u m b e r of essential ideas and techniques for the study of inverse problems that are ill posed. T h e r e is a clear emphasis on the mollification method and its multiple applications when implemented as a space marching algorithm. A s such, this book is intended to be an outline of the numerical results obtained with the xi

xii

PREFACE

mollification method and a manual of various other methods which are also used in arriving at some of these results. Although the presentation concentrates mostly on problems with origins in mechanical engineering, many of the ideas and methods can be easily applied to a broad class of situations. This b o o k — a n outgrowth of classes that the author has taught at the University of Cincinnati for several years in the Seminar of Applied Mathematics—is organized around a series of specific topics aimed at upper-level undergraduates and first-year graduate students in applied mathematics, the sciences, and engineering. It may b e used as a primary test for a course on computational methods for inverse ill-posed problems or as a reference work for professionals interested in modeling inverse p h e n o m e n a in general. T h e treatment is strongly computational, with many examples and exercises, and truly interdisciplinary. T h e r e is m o r e than enough material in the book to b e covered in a semester or two-quarters-long course. Although all the problems considered are physically motivated, a knowledge of the physics involved is not essential—but always very useful!—for the understanding of the mathematical aspects of these problems. It has been my experience, after teaching this material several times, that the subject is most appreciated by the students when they write programs of their own and see them work. This is the main reason for having computational exercises in the book. A n "experimental" approach to numerical modeling—denning and carefully testing the new numerical methods on simple problems with known solutions, before attempting to formally prove their stability and accuracy—is often the best way to proceed. T h e students are advised to work o n as many exercises as possible in each chapter, as this is the best way to learn the material and to check if students master the subject. Chapter 1 begins with the classical topic of numerical differentiation as an inverse problem. T h e mollification method is t h e n introduced a n d a thorough discussion of its numerical implementation follows. In Chapter 2 we investigate Abel's integral e q u a t i o n — a n o t h e r classical problem of mathematical physics—and four different methods are developed and placed in their p r o p e r computational framework. Chapter 3—the main thrust of the book—is devoted to the one-dimensional inverse heat conduction problem (IHCP). T h e discussion of several mathematical models and their pertinent numerical algorithms for the approximate determination of the unknown transient t e m p e r a t u r e and heat flux functions is developed here. Chapter 4 on the two-dimensional I H C P is presented as an integral part of the text. It is in this topic area where the r e a d e r can find the most prolific and challenging source of new problems. Chapter 5 contains t h r e e independent applications for space marching solutions of the I H C P : identification of boundary source functions and radiation laws, numerical solutions of the Stefan problem and inverse Stefan problem, and determination of the initial t e m p e r a t u r e distribution in a

PREFACE

xiii

one-dimensional conductor from transient measurements at interior locations. Chapter 6 illustrates further applications of stable numerical differentiation techniques ranging from the determination of forcing terms in systems of ordinary differential equations to the identification of transmissivity coefficients in linear and nonlinear elliptic and parabolic equations in one space dimension. Appendix A offers a selected overview of the essential mathematical tools used in these lectures. It should constitute a genuine aid for nonmathematicians. Finally, Appendix Β contains an up-to-date citing of the literature related to the I H C P . It might certainly be of value to graduate students and researchers interested in the subject. DIEGO A . MURIO

ACKNOWLEDGMENTS

I would like to acknowledge the contribution of several of my graduate students—Constance Roth, Lijia G u o , and Doris Hinestroza—who helped m e test and debug portions of the original manuscript u n d e r diverse circumstances. I feel particularly indebted to Carlos E. Mejia who read the entire text and m a d e many valuable suggestions while still working on his Ph. D . Dissertation. Finally, I wish to acknowledge the following organizations for permission to reproduce the indicated tables and figures: Society for Industrial and Applied Mathematics (SIAM), Figures 2.6 and 2.7; I O P Publishing Ltd., Figures 4.1a, 4.16, 4.2a, and 4.2b and Tables 4.1 and 4.2; Pergamon Press, Figures 5.1 to 5.7, 5.10 to 5.13, 6.3, 6.6, 6.76, and 6.8 and Tables 5.1, 5.2, and 6.1 to 6.3. D . A. M.

XV

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

1 NUMERICAL DIFFERENTIATION

In several practical contexts, it is sometimes necessary to estimate t h e derivative of a function whose values are given approximately at a finite n u m b e r of discrete points. It is easy to imagine many different situations— mostly involving integral equations and ordinary and partial differential equations—related with t h e question of numerical differentiation of measured (noisy) data. Several interesting applications of this basic problem will b e investigated in t h e following chapters.

1.1

DESCRIPTION OF THE PROBLEM

In order to gain some insight on the underlying principles, let us analyze first t h e ideal situation where we seek an approximation to t h e derivative function / ' ( * ) u n d e r the assumption that t h e exact (errorless) data function fix) is sufficiently smooth on a given interval [a, b]. For example, if we assume that / e C ( [ a , b)]—third derivative continuous on [a, b]—and satisfies t h e uniform b o u n d 3

ΙΙ/ΊΙ..Ι..6] -

max | / " ' ( * ) I £Af.3>

a<x/(*)

=

+

h)-f(x-h) 2h

by the centered difference

a

X η, set 8

max

u p d a t e d value of 8 is always given by {-(5

min

+

= 8. T h e

8 ). max

Step 5. R e t u r n to Step 2. Once t h e radius of mollifcation δ and t h e discrete function 7 - / a r e determined, we use centered differences to approximate t h e derivative of Jsf at t h e sample points of t h e interval K - = [38,1 - 3 δ ] . s

m

1.6

m

s

NUMERICAL RESULTS

In this section we discuss t h e implementation of the numerical m e t h o d and the tests which we have performed in order to investigate t h e accuracy and stability of t h e numerical differentiation procedure. In all t h e examples, h = 0.01, 8 = 0.1, and / = [0,1]. T h e exact data function is denoted by fix) and t h e noisy data function f ix) is obtained by adding an ε random error to fix), that is, max

m

/„(*/)

(1.8)

=fix )+s6 i

i

where JC, = ih, i = 0 , 1 , . . . , N, Nh = 1, and 0, is a uniform random variable with values in [ — 1,1] such that *,.) -/(*,)|

O 0.60-

Λ

*

> c

φ 0.46-1 Q 0.40H

0.36 Η

β· !—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—r-^f 0.30 0.40 0.50 0.60 0.70 38-

x-vaIues Fig. 1.3 fix) = x(l - \x)\ (* * *); exact (- - -).

ε = 0.02, 8 = 0.10, h = 0.01. Computed

derivative

NUMERICAL RESULTS

15

15.00-1

Λ

12.6010.00H


' 0.00 Η — ο

-2.50 H

>

L Φ -6.00H

Ω

#

/

ι

-7.50-10.00-H

ν

ν

• *

ν

-12.50H -16.00-(-η—| 0.00

| | | I—I—I I | I I I I | I I I I | 0.26 0.50 0.75 1.0(9 Χ- Vβ IUθS

Fig. 1.4 fix) = sin4irx; ε = 0.01, δ = 0.02, h = 0.01. Computed derivative (* * *); exact (—). derivative fix) = 1 - x. With ε = 0.05, t h e corresponding radius of mollification in this case is 5 = 0.2. T h e resolution in this problem is quite good considering t h e relative high noise level which we used. T h e associated error norms a r e given by e = 0.001108 a n d = 0.00576. l h

h

E X A M P L E 2 O u r second example is rather oscillatory o n [0,1]. W e choose fix) = sin47rjc a n d for ε = 0.01 we obtain δ = 0.02. In Fig. 1.4 we plot t h e numerical solution obtained with t h e mollification method ( * ) a n d t h e exact derivative fix) = 4 i r cos 4πχ. T h e corresponding relative error norms a r e given by ! EXAMPLE 3

2 , / i f

= 0.01976

and

!"'*

= 0.02439.

In Fig. 1.5 we plot t h e exact derivative of fix)

= sin I O t t *

and t h e solution obtained with t h e mollification method ( * ) . T h e data

16

NUMERICAL DIFFERENTIATION

0

i

11

ϋ 0.00Ο

>

-10.00H

Φ Q

I

1

τ

I

I

I

4

1

I

I

1 -20.00

ϊ

ι ι J

0.00

I

1 4

t

ι

W I

-30.00H

-40.00

Γ

ιι

h > 0, k integer, consider the grid points χ = kh in the whole discrete line and define the discrete—see Sec. A.4—averaging kernel 1 kh . Trexp kh Ph.s = \ s 2

c

2

2

2

δ

2

\kh\ < 8, \kh\ > 8,

18

NUMERICAL DIFFERENTIATION

where C = ηΣΊ exp[k h /(k h integer less than or equal to 8/h. 2

s

(a) hZ p k

2

2

- δ ) ] . H e r e Ν denotes the largest Prove that

2

2

Ν

= 1.

KS

(b) Given a discrete function

g, h

(D (pH,s*g ))(M) 0

where (f

h

=

h

* g Xkh)

{ , *(D g) )(kh), Ph B

= hZj(jh)g((k

h

1.2. {Discrete analogous of Lemma l | I > g L < M, show that

-

0

h

j)h).

1.1) With p

h

δ

as in Exercise 1.1, if

2

A

\D ( , *g ) 0 Ph s

-D g \l<SM

h

0

h

and \\D (Ph,>*8h)

-D g \l<SM.

+

1.3. With p

h

s

+

h

as in Exercise 1.1, show that if δ is a multiple of h, then £>o(Ph,s*S )

(D )*g .

=

h

oPhS

h

1.4. (Discrete analogous of Lemma 1.2) With p as in Exercise 1.1, if δ is a multiple of h and g is a noisy discrete function such that \\g g \U < e, show that h

h

s

m

h

Km

2ε \£>o(Ph,a* Sh, ) m

1.5. (Discrete error estimate) show that

h s

0

h a

£

h

3δ

U n d e r the conditions of Exercises 1.2 and 1.4,

\\D (p , *g , ) 0

~ D (p , *g )L

h m

2ε —.

-D g \l