Dynamic programming and partial differential equations, Volume 88 (Mathematics in Science and Engineering)

Dynamic Programming and Partial Differential Equations

This is Volume 88 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A partial list of the books in this series appears at the end of this volume. A cornplete listing is available from the Publisher upon request.

Dynamic Programming and Partial Diferential Equations E D W A R D ANGEL Department of Electrical Engineering University of Southern Colifornio Los Angeles, California

ACADEMIC PRESS

R I C H ARD BELLMAN Departments of Mathemotics ond Engineering University of Southern California Las Angeles, California

New York and London

1972

COPYRIGHT 0 1972, BY ACADEMIC PRESS,INC.

ALL RIGHTS RESERVED NO PART O F THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New

York 10003

United Kingdom Edition published b y ACADEMIC PRESS, INC. (LONDON) LTD.

24/28 Oval Road, London N W l IDD

LIBRARY OF CONGRESS CATALOG CARDNUMBER:70- 189674 AMS(M0S) 1970 Subject Classifications: 35560, 35K55 PRINTED IN THE UNITED STATES O F AMERICA

Contents

xi

PREFACE CHAPTER 1

INTRODUCTION

1

CHAPTER 2

QUADRATIC VARIATIONAL PROBLEMS 1. 2. 3. 4. 5. 6. 7.

Introduction Variational Approach Positive Definiteness, Existence, and Uniqueness of Solution Computational Aspects Vector-Matrix Case Rayleigh-Ritz Method Bubnov-Galerkin Method Bibliography and Comment

6 7

8 8 9

10 10

11 V

vi

Contents CHAPTER 3

D Y N A M I C PROGRAMMING 12 12 13 14 14 15 17 17 18 19 19 20 21

Introduction Difference Equations Functional Equation Principle of Optimality Nonstationary Case Quadratic Functions Minimum Convolution Acceleration of Calculation Differential Equations Quadratic Case 1 1 . Minimum Convolutions 12. Tridiagonal Matrices Bibliography and Comment

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

CHAPTER 4

T H E POTENTIAL E Q U A T I O N 1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

II. 12. 13. 14. 15. 16.

17.

Introduction The Euler-Langrange Equation Inhomogeneous and Nonlinear Cases Grcen’s Function One-Dimensional Case Two-Dimensional Case Discrctization Rectangular Region Rigorous Aspects Associatcd Minimization Problem Approximation from Above Discussion Semidiscretization Irregular Grid Solution of the Difference Equations Itera!ive Solutions Limitations of the Iterative Approach M iscellancous Exercises Bibliography and Comment

22 23 24 24 25 27 28 28 29 29 30 30 30 31 32 32 34 34 35

CHAPTER 5

D Y N A M I C PROGRAMMING A N D ELLIPTIC E Q U A T I O N S I . The Potential Equation 2. Discretization

37 38

Contents

3. 4. 5. 6.

I. 8.

9.

10. 11. 12. 13. 14. 15. 16.

vii 39

Matrix-Vector Formulation Dynamic Programming Recurrence Equations The Calculations Nonsingularity Stability Discussion Efficiency Example Deferred Passage to the Limit General Linear Equations Irregular Regions Higher Order Equations Distributed Control Bibliography and Comment

40 41 42 43 44 46 47 48 49 50 52 55 57 59 CHAPTER 6

I N V A R I A N T IMBEDDING I. 2.

3.

4. 5. 6.

1. 8.

9.

10. 11. 12.

13.

14. 15.

16.

Invariant Imbedding The Riccati Transformation Single Sweep Methods Discretization Recurrence Relations Relation to Dynamic Programming Nonsingularity and Stability Relation to Gaussian Elimination Relation to the Riccati Equation Invariant Imbedding Continuous Invariant Imbedding Generalized Riccati Transformations The Biharmonic Equation Random Walk Invariant Imbedding and Random Walk Another Imbedding Bibliography and Comment

61 62

64 67 68 69 69 71 71 72 75 78 80 82

83 84 85

CHAPTER 7

IRREGULAR REGIONS Introduction Irregular Regions 3. Case I: Order uR > Order uR-, 4. Example 5 . Case 11: Order uR < Order UR-1 1

2.

89 89 90

92 93

viii

Contents

6. Example 7. Nonsingularity and Stability 8. Rcmoval of Restrictions 9. Examples 10. General Linear Equations I I . Other Boundary Conditions 12. Three Dimensional Equations 13. The Biharmonic Equation 14. Invariant Imbedding and Difference Equations 15. A Second Approach 16. Matrix-Vector Equations 17. General Regions Bibliography and Comment

94 95

96 97 98 99

101 103 104 109 111 1 I4 1 I7

CHAPTER 8

SPECIAL CO MP U T A T l 0 NAL MET H ODS 1. 2.

3. 4. 5. 6. 7.

8. 9. 10. II. 12.

Direct versus Iterative Methods The Characteristic Values of Q Kronccker Product Kronecker Sunis An Example Another Direct Method Diagonal Decomposition Point Iterative Methods The Successive Overrelaxation Method Block Iterative Methods A I ter nat ing-Direct ion I nip I icit Met hods Discussion Bibliography and Commcnt

1 I9 120 121 122 123

125 126 128 130 132 134

135 136

CHAPTER 9

U N C O N V E N T I O N A L DIFFERENCE M E T H O D S I. Introduction 2. Invariant Imbedding 3 . The Equation / l , = i u i X 4. Approximating Finite Difference Equation 5. Convergence 6 . Improvement of Accuracy

138 138 139 139

140 141

Contents 7.

IX

i42 143

Differential Quadrature Bibliography and Comment CHAPTER 1 0

PARAB0LIC E Q UAT10 NS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

The Heat Equation Properly Posed Problems Consistency and Stability Explicit Methods Implicit Methods Crank-Nicholson Method Alternating-Direction Implicit Methods The Laplace Transform Gaussian Quadrature Inversion of the Laplace Transform Computational Aspects Bibliography and Comment

144 146 147 148 151 152 153 156 157 158 160 161

CHAPTER 11

N O N L l NEAR EQUATIONS A N D QUASlLl NEARlZATlON 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Introduction Successive Approximations Quasilinearization An Example The Equation u,,+uyy = uz A Differential Inequality Monotonicity Maximum Domain of Convergence Quadratic Convergence Computational Aspects Example Identification Problems The Least-Squares Criterion Newton-Raphson-Kantorovich Method The Sensitivity Equations Quasilinearization Example Miscellaneous Exercises Bibliography and Comment

162 163 163 164 165 166 167 169 169 170 171 172 172 173 174 175 176 179 179

Contents

X

APPENDIX

COMPUTER PROGRAMS Program Program Program Program

1. Dynamic Programming 2. Riccati Transformation 3. Invariant Imbedding 4. Quasilinearization

AUTHORINDEX SUBJECT INDEX

182 186 190 194 20 1 203

Preface

A major task of contemporary mathematical analysis is that of providing computer algorithms for the numerical solution of partial differential equations of all types. This task requires a motley mixture of methods. In this book we wish to show first that dynamic programming and invariant imbedding furnish some powerful techniques for the solution of linear elliptic and parabolic partial differential equations over regular and irregular regions. Nonlinear equations are made accessible by the use of quasilinearization which can be combined with the earlier procedures to study identification and inverse problems. What is quite important is that these algorithms are straightforward, easily learned, readily programmed, and easily used. We wish to thank John Casti, Dave Collins, Nestor Distefano, and Art Lew for much helpful advice and comments in the preparation of the manuscript. We are particularly grateful to John Todd of the California Institute of Technology who read the manuscript through in his usual painstaking fashion and made a large number of suggestions based on his vast erudition and considerable experience in this field which materially improved both content and presentation. Edward Angel was with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley during the preparation of the manuscript. xi

This page intentionally left blank

Chapter I

lntroduction

The principal aim of this volume is to apply some of the ideas and methods of the theory of dynamic programming to the analysis and computational solution of the potential equation uxx

+

uyy

= 0Y

(1)

with the boundary condition u(x,.J4 = d X , Y ) ,

(X,Y)Er,

(2)

where r is the boundary of a region R which may be of quite irregular shape as in Fig. 1.

FIGURE 1

P

1

2

1 Introduction

There are currently a large number of methods available for handling questions of this nature, all of which work well in carefully chosen circumstances. All, however, possess various drawbacks, including, of course, the methods presented below. It is this fact which motivates a constant search for new methods. The pros and cons of these techniques are discussed in appropriate places. Variations on the principal method and the principal equation lead to other methods and equations of interest in their own right. We shall discuss these below. Let us begin with the basic idea. I n variational problems involving functions of one space variable, say the minimization of

J(u) =

sur

g(U, u’)dt ,

(3)

subject to ~ ( 0= ) c, a fundamental observation is that the minimum value of J ( u ) depends upon the quantities c and T. We then regard c and T as variables, with T 0 and - 00 < c < co,and write

f ( c , T ) = min U J(u) .

(4)

The original variational problem is thus transformed into that of obtaining an equation for f’(c,T). This is the approach of the theory of dynamic programming: we imbed a particular problem, with fixed values of c and T, in a family of problems in which c and T assume ranges of values. Then we obtain relations connecting various members of the family. An equation forf(c, T ) is obtained by viewing the minimization process as a multistage decision process in which a choice of a function over [O,T] is decomposed into a choice over the subinterval [0,s] and then a choice over the subinterval [s, T I , 0 < s < T. Thus min = min min ul0,TI

ulO,aIu[5,Tl

(5)

This is readily translated into a functional equation for f ( c , T ) using the principle of optimality. If we let s+O, this equation becomes a partial differential equation. This is discussed in Chapter 3. A similar approach can be pursued in the multidimensional case. We begin with the fact that the partial differential equation in ( I ) is the EulerLagrange equation associated with the quadratic functional,

1 Introduction

3

the Dirichlet functional. As above, we can regard the minimization of J ( u ) as a multistage decision process in which the choice of a function over R is decomposed into a choice over a subregion R , and then over the region R - R , . This yields min ueR

=

min min . u t R , u t ( R- R , )

(7)

Different choices of R , yield a variety of interesting and useful results. Thus, for example, if the region R is rectangular, we can choose R , as a rectangular strip (Fig. 2). This leads to either a recurrence relation or a

FIGURE 2

FIGURE 3

differential equation depending upon the type of discretization we employ and whether R , is taken to be infinitesimal or not. If the region has the form shown in Fig. 3, we choose R , as indicated in the figure and the equation over the region R = R , + R , can be solved in terms of its solutions over the simpler regions R , and R,. All of this will be discussed in some detail in subsequent chapters, and numerical applications will be given. We shall not, however, discuss the corresponding problems and procedures for circular and spherical regions. Let us now examine the contents of individual chapters. In Chapter 2 we review the minimization of the quadratic functional

J(x)

=

[W.

x')

+ (x,A (t>x)l dt ,

(8)

using the classical approach of the calculus of variations. This leads to the Euler-Lagrange equation

x" - A ( t ) x

=0 ,

(9)

subject to various boundary conditions, depending on the conditions originally imposed on x. Some of the problems encountered in a numerical solution are discussed there. At the close of the chapter there is a brief sketch of the Rayleigh-Ritz and Bubnov-Galerkin methods. These are

4

1 Introduction

powerful techniques for treating the linear partial differential equations of mathematical physics and engineering. The drawback is that they lead to the task of solving large systems of linear algebraic equations. Our concern with questions of this nature is due to the fact that discretizations of various types transform the Dirichlet functional into a functional of the foregoing type and thus to equations of the type appearing in (9). I n Chapter 3 we present the dynamic programming approach to the minimization of J ( x ) as well as to the minimization of its discrete version

I n this way we encounter the Riccati equation and its discrete analogue. Chapter 4 is devoted to the potential equation, Green’s functions, and various discretization procedures. Chapter 5 contains a more detailed discussion of the application of dynamic programming to the discrete case. Chapter 6 describes some applications of the theory of invariant imbedding to the solution oflinear partial differential equations. An advantage of this technique is that we can apply it to general linear equations, not necessarily derived from a variational principle. In Chapter 7 we discuss the details of the application of the foregoing methods to irregular regions, while Chapter 8 is devoted to some special methods which can be used for regular regions. The theory of dynamic programming yields some nonlinear partial difrerential equations describing continuous decision processes which are conveniently resolved computationally using the analogous equations for the associated discrete decision processes. This suggests a corresponding procedure for other types of partial differential equations. To illustrate this idea, in Chapter 9 we examine the use of the recurrence relation

to study the equation

and some associated questions. The parabolic equation

1 Introduction

5

the heat equation, is studied in Chapter 10 using the Laplace transform to reduce it to an equation of elliptic type which is then studied by the foregoing methods. Numerical inversion of the Laplace transform then yields values for u. Finally in Chapter 11 we examine the use of the methods of successive approximations, using quasilinearization, to resolve nonlinear equations. Throughout we are concerned both with the analysis itself which is extremely interesting and with the equally interesting question of providing numerical answers to numerical questions using reasonably simple straightforward methods and easily written computer programs. Our constant theme is that these two types of problems are closely interrelated. The analysis must be judged o n whether or not it is computationally operational, while the derivation of operational methods often require new analytic procedures.

Chapter 2

Q u a d r a t i c Var i a t i on a I Pro b le m s

1. Introduction In this chapter we wish to review some of the more important classical methods for treating linear differential equations and quadratic variational problems. An understanding of what is required will motivate o u r subsequent application of a different approach, dynamic programming to the same problems. At the outset however, it should be explicitly pointed out that all methods race dilficulties in treating problems of high dimension. These difficulties are different for different methods. We shall first discuss the minimization o f the scalar quadratic functional

J(u)

(u”

= JO’

+ q(t)U2)dt,

subject to u ( 0 ) = c , , u ( T ) = cz and then the N-dimensional counterpart, the problem of minimizing

J(x) = 6

I’

[(x’, x’)

+ (x, A ( t )x)] dt ,

(2)

7

2. Variational Approach

subject to x(O)=c, x ( T ) = d . Our tool will be the associated EulerLagrange equation. Following this, we shall briefly sketch the Rayleigh-Ritz and BubnovGalerkin methods. All three procedures ultimately require the solution of linear algebraic equations, and the obstacles stem from this.

2. Variational Approach Let us suppose that there is a function u such that u and u' belong to Lz (0, T ) , u satisfies the boundary conditions, and u furnishes the absolute minimum of J(u). Our aim is to obtain a necessary condition that u must satisfy. This is the Euler-Lagrange equation which in this case turns out to be sufficient. We proceed as follows. Let u be another function such that u and u' belong to L'(0, T ) with v(0) = u(T) = 0. Then for a n y real E , the function u+ EU satisfies the original boundary conditions and is such that it and its derivative are in L2(0,T ) . Consider next the expression J ( U + E U )= J ( u )

+ c2J(u) + 2~

[u'u'+q(r)u~]dt,

which by assumption possesses an absolute minimum at yields the variational condition

E = 0.

(1)

This fact

for all u of the foregoing nature. Integrating by parts, this becomes

+

u l q ( t l ) u ( t , ) d t l ]0 T / T 0 [ u ' u ' - ~ r/ f0q ( t , ) u ( t , ) d t ,

The integrated term drops out since v ( 0 ) = u ( T ) = 0. Hence, for any constant c j , c u t [c3

+ u' - ~ q ( r l ) u ( r l ) d r l dr, ] = 0.

(4)

Choose q(tl)u(r,) dt,

v(0) = 0 ,

(5)

8

2 Quadratic Variational Problems

where c j is determined by the condition v ( T ) = O . Then (4) yields the relation c3

+ u’ -

J6’4(f)’

df, = 0

u(f1)

almost everywhere. The almost everywhere may be replaced by everywhere. Thus (6) yields the Euler-Lagrange equation u” - q ( t ) u = 0 ,

u(T) =

u(0) = c , ,

c2.

(7)

3. Positive Definiteness, Existence and Uniqueness of Solution It is not difficult to show that if J ( u ) is positive definite, then (2.7) possesses a unique solution. We proceed as follows. If there are two solutions, there is a solution u such that ~ ( 0=) v(T)= 0. Consider then the expression 0

J’,

T

=

T

V(V”

=

0-

L

T

- q ( f ) v ) d f= v z ~ ’ ]-~

L

T

[v’2

[ d 2+ q ( t ) v 2 ] d f

+ q ( t )V 2 ] df,

(1)

a contradiction to the assumed positive-definite character of J ( v ) if u is not identically zero. That (2.7) possesses a solution follows from the discussion below.

4. Computational Aspects Let u1 and u2 be the principal solutions of (2.7), i.e., U ’ ( 0 ) = 1,

u,(O) = 0 ,

u; (0) = 0 ,

ui(0) = 1 .

Write u = a, u ,

+ a,u,,

(1)

(2)

where a , and a , are to be determined by the conditions (‘1

=

a,,

c2

=

a , UI ( T )

+ a, u,(T).

(3)

The positive-definite nature of J ( u ) guarantees that u 2 ( T ) # 0, as indicated in Section 3, whence a , and a, are uniquely determined. The values of the principal solutions are determined by numerical integration of the differential equation (2.7) with the appropriate initial conditions.

9

5. Vector-Matrix Case

5. Vector-Matrix Case Let us consider in similar fashion the task of minimizing

J(x)

=

/)(xf,x’)

+ (x,=4(t)x>3d t ,

where A ( t ) is positive definite. We suppose that x(0) = c, x(T)= d and x’~L’(0,T ) . Then proceeding as before the variational equation is X”

x ( 0 ) = c , x(T)= d .

-A(t)x = 0,

(2)

Analogous arguments to those given in Section 3 show that this equation possesses a unique solution. The computational aspects, however, require a careful examination. As a straightforward extension of the method used in the scalar case, write where X , and X 2 are the principal matrix solutions of

X ” - A ( t ) X = 0,

(4)

that is,

X,(O)

=

I,

Xi(0)

X,(O)

=

0,

Xi(0) = I ,

=

0,

and a and b are constant vectors to be determined by the boundary conditions given in (2). These lead to the equations c =

a,

d

=

X,(T)a+ X2(T)b.

(6)

It is not difficult to extend the method sketched in Section 2 to the present case to show that X , ( T ) is not singular. Hence a and 6 , and thus x ( t ) , are uniquely determined by (2). This determination of b, however, requires a solution of a system of linear algebraic equations, always a ticklish matter. It is particularly so when the dimension of a is large and Tis large. The fact that T is large means that X 2 ( T )is close to a singular matrix. This combined with the fact that dim(X2) is large means that numerical accuracy is not readily ensured. A detailed discussion of these matters will be found in the reference cited in the bibliography at the end of the chapter. I t is the block to a straightforward solution of linear differential equations of high dimension that motivates the constant search for new approaches to linear partial differential equations and quadratic variational problems.

10

2 Quadratic Variational Problems

6. Rayleigh-Ritz Method

Let us briefly sketch two of the most powerful approaches which avoid the route indicated i n Section 5. The first is the Rayleigh-Ritz method. Consider the functional J(x) =

lr

[(x', x')

+ (x,A ( t ) x)] d t ,

with x(0) = c, x ( T ) = d, and use the trial function

We have a choice of letting the 4k be scalars and the 6, vectors or conversely. In any case, let us choose the & as known functions and the b, as unknowns, subject only to the boundary conditions. Then J ( x ) = J @ , , b,, ..', bM)

(3)

and the minimization is now over the 6,. This leads to a set of linear algebraic equations, of degree M if the bi are scalars. If M N , this is a much more tractable problem than the original. If we do not wish to use the boundary conditions, we can form the new functional

J(x, I . 1 , & )

=

J(x) + AI(x(0)- c,x(O)

- C)

+ A ~ ( x ( T-) d , x ( T ) - d ) ,

(4) where I., and A2 are Courant parameters and A, and A2 % 1 . The choice of the sequence { 4 , ( f ) }is critical and is determined in any particular case by some combination of mathematical reasoning, physical intuition, and experience.

7. Bubnov-Galerkin Method I n place of solving the Euler-Lagrange equation x"- A ( t ) x =

0,

x(0) =

C,

x(T) = d ,

consider the problem of minimizing PT

J,(x) =

J -(x"-A(t)x,x"0

A(t)x)dt

11

Bibliography and Comment

over x ( t ) such that the boundary conditions are fulfilled and the integral exists. Now use the Rayleigh-Ritz method to carry out this minimization. It is sometimes convenient t o introduce the mixed expression J , (x) = J(x)

+ AJl (x)

(3)

in place of J(x) alone, where I again is a Courant parameter. BIBLIOGRAPHY A N D COMMENT Sections 1-5. A detailed discussion of these problems plus a number of additional references may be found in

R. Bellman, Introduction

to the Mathernutical Theory of Control Processq I : Linear Equations and Quadratic Critwia, Academic Press, New York, 1967.

For an introduction t o the regularization techniques of Tychonov, see

R. Bellman, R. Kalaba, and J . Lockett, Nunwrical Inriersion Amer. Elsevier, New York, 1966. Sections 6-7.

of the Laplace Transfimn.

See

R. Bellman, Methods of Nonlinear .4nalvsis, Vol. I , Academic Press, New York, 1970.

Chapter 3

Dynamic P rograrnrning

1. Introduction In this chapter we wish to describe some applications of the theory of dynamic programming to variational processes governed by either differential or difference equations. The techniques yield particularly elegant results when applied to quadratic functionals associated with linear equations of state. Furthermore, and fortunately, these are the functionals and equations met in a variety of situations. Utilization of these results for numerical purposes will be discussed in the following chapters.

2. Difference Equations Let x,,, ti = 0, I , 2, ... be a K-dimensional vector, the state vector, determined recurrently by the difference equation X"+1

= dX,,,Y"),

xo = c ,

(1)

where y,, is an L.-dimensional vector, the control vector. I n many situations L = K. We suppose the set of control vectors, {y,}, is to be chosen to 12

3. Functional Equation

13

minimize the criterion function (or “return” function),

This defines a deterministic control process of discrete type. Assuming that the problem is well-posed in the sense that the minimum is achieved, as it is easy to show in the cases of interest to us in what follows, we set

a function defined for all c, and N

= 0,

1,2, ... . We see that

a function which in many cases is readily determined. This is a fundamental imbedding, an idea we shall discuss in more detail in the chapter on invariant imbedding. I n order to resolve a particular problem where c and N are specified, we proceed to obtain a functional equation forf,(c) considered as a function of c and N . 3. Functional Equation As noted above, the functionf,(c) is given by f o ( c ) = min h(c,y) . Y

(1)

I n view of this determination, we would like to obtain a recurrence relation, another difference equation, which connects fN+ ( c ) with f i ( c ) . Theoretically this would lead to a constructive way of obtaining the sequence f , ( c ) . Operationally, as we shall see, there are some additional points that remain. The form of the criterion function in (2.2), its additivity, allows this relation to be readily obtained. We have f+I(C)

min

=

J ( { x , ) , {Y,>)

(Y.1

n=0,1,2 ,_.., N+l

=

min Y”

min {Y“ )

n = 1,2, ..., N + 1

J ( { x , ) , {y,})

14

3 Dynamic Programming

Since so= c, and x1 = g(x,, y o ) = g ( c , yo), this yields the quantity

upon recalling the definition of { f N ( c ) } . Since y o is a dummy variable here, we can eliminate the subscript and use the simpler form

This is the fundamental recurrence relation which determines both the sequence {fN ( c ) }and the control variables which minimize.

4. Principle of Optirnality The functional equation in (3.3) becomes immediate when we conceive of the process as a multistage decision process and apply the principle of optimality of the theory of dynamic programming. Principle of Optimality. A n optimal policy has the property that whatever thc initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting f r o m the$rst decision. Frequently the applicability of the principle is immediate, as is a proof by contradiction. This is, for example, the situation in the foregoing problem. In all cases, however, it is necessary to validate its use.

5. Nonstationary Case In many cases of importance the functions appearing in the description of the criterion function are time-dependent which means that J has the form

where the single-stage return depends upon the stage. T o handle problems of this nature using the same concepts, we reverse the direction of time and introduce the function

15

6. Quadratic Functions

defined for k = 0,1,2, ..., N , - co < c < 00. Here N is fixed and k, the starting time, is variable. The function 4 N ( ~ is )now readily obtained, namely

4 N (c) = min

(3)

(c, y>9

and the desired recurrence relation is

4 k ( C ) =min[h,(c,y)+4k+,(g(c,y))],

k

=

0 , 1 , . . . , N - - 1 . (4)

6. Quadratic Functions The functional equations can be considerably simplified in the case where the criterion function is quadratic and the state equation is linear. Consider, for example, the problem of minimizing the quadratic expression

where x and y are related by the equation X,+l

=

x,+ cy,,

xg =

(2)

c.*

Let us suppose that A and B are positive definite so that the minimum exists. Writing as before fN(c)

=

min J ( { x n ) , {Y" )

a function defined for - co < c < co, N (3.31, f N + 1 (c>

=

min [(c, Ac) Y

+ (y,

bn>)

(3)

9

= 0,1,2, . ..

+fN(c

+ cy)]

. We have, =

9

O,

following

...) (4)

with

f ( 4 = min C(c,Ac) + (Y,BY)l

= (C,AC)

.

(5)

It remains to simplify (4) and thus obtain some relations of more tractable analytic and computational nature. To this end we make strong use *We are employing the usual inner product representation of quadratic forms. Thus (a,b) = C;"=a& where aiand bi are the ith components of a and b respectively, whence ( x , A x ) = Z;", j = l a,,xixj. We use xI,xZ,~~~,y1,y2, ... to denote a sequence of vectors above to avoid the necessity of a superscript notation.

16

3 Dynamic Programming

of the fact thatf,(c) is quadratic function of c,

where the matrix QN is independent of c. The quadratic nature offN(c) is apparent from the linear structure of the usual variational equations. In any case it may readily be established inductively using (4) as the following argument indicates. Using (4) and (6), we have

The point is that the minimum on the right hand side, as well as the minimizing value of y , can readily be determined. The minimizing y is a linear function of c. Thus, the evaluation of the expression will yield another quadratic form in c. Equating coefficients yields a recurrence relation connecting Q N + with Q N .We shall carry through the details in the following chapter in connection with the variational problem of particular interest to us.

,

EXERCISES

,

+

1. Consider the problem of minimizing J({u,,}) = Cf=’=, [(u,, - u,J2 an2], where uo = c‘. Show by a change of variable that fN(c) = min J = rNc2. U

2. Using the procedure indicated above, obtain a recurrence relation connecting r, with r,,

3. Show in this way that limN+mr, exists and determine its value. 4. Show directly that rN< r N + ,, that rN is uniformly bounded, and hence that rNconverges as N + co. 5. Consider the functionf(c) = minuC,7=,[un2+(u,-u,- ,)2], uo = c. Show that , f ( c ) = min [c’+(a-c)* +f’(u)] and hence determine roo directly. I,

6. lff’(0) = 0 andf(c) is analytic in c, does the functional equation determine f’( c) uniquely ?

7. Show that min [(x, A.u)-2(x,y)] X

=

-(y, A-’y).

8. Acceleration of Calculation

17

7. Minimum Convolution The same ideas may be employed to obtain some more general results. Introduce the function of two variables

where x, is subject to two constraints x0 = C,

XN

= d7

(2)

and x and y are related by the relation Write 4M+N(C,d)

=

min

=

min

M+N

[c

n=O

n=M+l

(4)

Let us choose a value for x M ,say xM = z. Then (2) yields

EXERCISES

1. What modifications are necessary if the single-stage return g(x, y ) , depends on n?

2. Consider the quadratic case and use (5) to derive a recurrence relation for r M + Ni n terms of r, and r,, where rN is defined as in the exercises at the end of Section 6. 8. Acceleration of Calculation In many cases we wish to determinefi(c) or 4 N ( for ~ )a particular value of N only. If N is large, it may be much more efficient to use the relations in Section 7 rather than relation (3.3). Thus, for example, (7.5) yields 42N+I(C)d)

=

min [ 4 2 N ( C , Z )

providing a relatively rapid determination of

+ 42N(Zyd)], cjl,

4 2 ,+ 4 , &, ... .

(1)

18

3 Dynamic Programming

To calculate fN(c), we can use the relation fN+M(C)

and thus obtain f 2 N

+I

=

min C 4 M ( C , 4 +fN(Z)l

in terms of f 2 N and

Y

42N.

EXERCISE

Obtain the corresponding recurrence relations when

f N ( c )= minJ(u) . U

9. Differential Equations Let us consider the problem of minimizing the functional

where x and y are connected by the differential equations x‘

= g(x,y),

(2)

x(0) = c .

Once again we suppose that the problem is properly formulated and set minJ(x,y) = f ( c , T ) .

(3)

Y

Write

SoT=SOS+l

(4)

Supposing that S is small, we obtain the approximate functional equation J’(c,T)= min[h(c,z)S+f(c+ z

Sg(c,z),T- S ) ]

+ o(s’),

(5)

where z = y(0). Passing to the limit as S+O, we obtain the nonlinear partial differential equation Jr

=

min [ h (c, 4

with the initial conditionJ’(c,O)

= 0.

+ ( 9 k z), gradf 11

9

(6)

19

11. M i n i m u m Convolutions

10. Quadratic Case The results simplify greatly when h ( x , y ) is quadratic in x and y and g(x, y ) is linear. Consider, for example, the minimization of PT

J ( x ) = J [(x’,x’) 0

+ ( x , A X ) ] dt ,

(1)

where A is positive definite and x(0) = c. It is clear that f ( c , T ) = minJ(x) X

=

(2)

(c,R(T)c),

where R ( T ) depends only on T. On the other hand, (9.6) yields fT

=

min z [(z,z )

+ (c, A c ) + ( z , gradf’)] .

(3)

The minimization with respect to z is readily accomplished, yielding

z = -gradf/2, fT =

(4)

(c, A c) - [(gradj; grad f I/ 41 .

(5)

Using (2) in (5), we obtain an ordinary differential equation, a Riccati equation, R’(T) = A - R 2 ( T ) ,

R(0) = 0 .

(6)

Using the value obtained above, we have

x’(0) = z

= - R(T)c,

(7)

the desired initial condition. 11. Minimum Convolutions

Write min /sTh(u.u’ t ) dt

= f ( a , b ;S , T ) ,

where u is subject to u ( S ) = a, u ( T ) = b. Then, as above,

f ( a , 6 ; S, T ) = min [f ( a , c ; S, R ) +f ( c , b ; R, T ) ] , C

S

< R < T.

(2)

20

3 Dynamic Programming

If h is quadratic i n u and u', thenfis quadratic in a and b.

+

f(a,b; S , T ) = a Z r l l ( S , T ) 2ahr12(S,T)

+ b2rZ2(S,T).

(3)

Using (2), functional equations for the r i j ( S ,T ) are obtained. EXERCISE

Obtain these functional equations. 12. Tridiagonal Matrices Let us consider the problem of solving the linear system AX

(1)

= C,

where A is symmetric and tridiagonal, i.e., l i - j l > 1.

a.. 11 = 0 ,

(2)

If we assume that A is positive definite, then the solution of (1) is equivalent to that of minimizing the quadratic form Q(x) = (x, A X ) - 2 ( ~X),

.

(3)

Let us write mi and /Ii to denote a. = I

pi

0..

11 7

= a,,,+ 1 =

Ui,,- 1

.

(4)

Now we can write out ( 3 ) as Q(x) =

~~,xI~+~/I~xIxZ+C~~X~~+~/I~X~X~+... + 2p,_ 1 x, - 1 x, + a, X"2 - 2c, x, - 2c2 X2 ". - 2cnX" . ( 5 )

Let us define Q,(x, z ) by Qk(X,Z)

= Mlxlz +2/IlX,x,+

+ xkxk2

Finally we define

Clearly since

-

2Cl

XI

'"+2/Ik-,x,-1Xk

- 2c2-Y~... - 2 z x k ,

k = 1,2, ..., . (6)

21

Bibliography and Comment

f.(c,) gives the minimum of Q(x). Applying the techniques we have just presented, it is easy to verify that

fk(’)

=

min [@k x k 2 - 2zxk X*

+fk- 1 ( c k - 1 + P k - x k ) l 1

9

(9)

with

fi(4 = z2/.1.

(10)

Furthermore, we can establish thatf,(z) is quadratic in z, fk(z) = rkz2

+ 2sk.2 + tk,

(1 1)

which enables us to find recurrence equations for the coefficients r k , s, and tk. Much of what follows in the succeeding chapters will be based on matrix versions of these results. EXERCISES

1. Prove (9).

2. Find the recurrence equations for the coefficients r k , s, and t k . Use these results to obtain a recurrence equation for the x k . BlBLlOG RAPHY AND COMMENT

See R. Bellman, Introduction to the Mathematical Theory of Control Processes I : Linear Equations and Quadratic Criteria, Academic Press, New York, 1967. Section 7 . See

R. Bellman, “Functional Equations in the Theory of Dynamic Programming XVII : Minimum Convolutions and Green’s Functions,” J. Math. Anal. Appl., Vol. 33, 1971, pp. 497499. Section 8. The problem of computing 2‘ with the least number of multiplications is an unsolved problem of mathematics. For some partial results, see

H. Kato, “On Addition Chains,” Ph.D. Thesis, Univ. of Southern California, 1970. D. E. Knuth, The Art of Computer Programming 11: Semi-Numerical Algorithms, Addison-Wesley, Reading, Massachusetts, 1969. Section 12. See

R. Bellman, Introduction to Matrix Analysis, 2nd ed., McGraw-Hill, New York, 1970. R. S. Lehman, “Dynamic Programming and Gaussian Elimination,” J . Math. Anal. Appl. Vol. 5, 1962, pp. 499-501.

Chapter 4

The P o t e n t i a l E q u a t i o n

1. Introduction In this chapter we wish to discuss some aspects of the potential equation, u,,

+ uyy = 0 ,

* = L7(X,Y),

k Y )5 R,

(1)

(-X,Y) E r,

(2)

where I- is the boundary of the region R in Fig. 1. We want to consider its connection with the minimization of the quadratic functional

,.

a ~ ( u= )

22

J

R

(u,’

+ uY2)d R ,

FIGURE I

(3)

23

2. The Euler-Lagrange Equation

the Dirichlet functional, and a number of problems associated i n this fashion. This quadratic minimization problem can be approached directly and effectively in many cases by means of Rayleigh-Ritz procedures, thereby furnishing an indirect approach to the potential equation. We shall follow a different route based on the use of discretization, a route which has advantages in certain cases. We will also be concerned with the Green’s functions associated with the inhomogeneous equation

and the more general equation

2. The Euler-Lagrange Equation We can readily show that the potential equation is the variational equation associated with D(u). Let u = 0 on r so that the functions u and u+ u satisfy the same boundary conditions on r. Then D ( u + u)

=

D(u)

+ D(u) + 2

SR

(u,u,

+ u,uJ

dR.

(1)

Applying Green’s theorem, we see that the third term vanishes if u satisfies the potential equation. Hence if u satisfies (1.1) we have

for any nontrivial u. Thus, if (1.1) and (1.2) possess a solution, they possess a unique solution, since any solution minimizes D ( u ) . Conversely, it can be shown from first principles that D ( u ) possesses a minimum over the class of functions u such that u,, u,€L2 with u satisfying the specified boundary condition. Further it can be shown that the minimizing function is determined by (1.1) and (1.2). Thus the problem of solving the potential equation and minimizing the Dirichlet functional D ( u ) are equivalent. We shall focus henceforth on the minimization problem.

24

4 The Potential Equation

3. Inhomogeneous and Nonlinear Cases I t follows from the foregoing that the problem of solving the inhomogeneous problem

u,,

+ uyy = W , Y ) ,

u(x,J4 = 0 ,

(X,Y) E

l- 3

(1)

and minimizing D,(y)

=

/R(u.2

+ u: + 2 h ( x , y ) u ) d R ,

(2)

subject to u = 0 on r, u,, uYeLZ(R), are equivalent. Similarly, the problem of solving

D,(u) = J (u,’ R

+ uy2 - k ( x ,y ) u’)

dR ,

(4)

are equivalent provided that D,(u) is a positive-definite functional. A sufficient condition for this is maxR I k ( x , y ) I < A , , where A, is the smallest characteristic value of the problem U , , + ~ ~ , + A ~= ~ 0,

u

=

0,

(X,~)EI-.

Finally, the non linear equation

u,,

+ uyy- h(u) = 0 ,

may he associated with the functional

where g’(u) = /7(u). We shall discuss this further i n the chapter on quasilinearization.

4. Green’s Function The solution of the inhomogeneous problem

5. One-Dimensional Case

25

may be expressed in the form 24.

= SRk(x,y,x,,l.,)h(r~,L.l)dR.

(2)

The kernel k is called the Green’s function associated with the particular equation and boundary condition. For the foregoing case we have k < 0, a result we wish to use subsequently. Let us derive this fundamental property directly from the associated variational problem.

5. One-Dimensional Case To illustrate the method of proof most simply, let us begin with the one-dimensional case. We wish to show that the solution of U”

+g(t)u = h(t),

u(0) = u ( T ) = 0 ,

(1)

is nonnegative in [0,T] whenever h ( t ) is negative, provided that the quadratic functional

is positive definite for functions satisfying the boundary conditions above. It follows from this hypothesis that ( I ) is the Euler equation associated with the minimization of

J,(u) = [ [ u ”

-g(t)u*

+ 2h(t)u]d t .

(3)

The solution of ( I ) furnishes the absolute minimum of ( 3 ) over the class of functions such that u’~L’(0,T ) , and the boundary conditions are satisfied. Let us suppose that the stated result is not true, which means that u, a continuous function, is negative in some interval [ a , b ] ; see Fig. 2. Here a may be 0 or b may be T. Consider the new function u defined by

The new function possibly has a discontinuous derivative at t = a and t = b, but nevertheless is such that U ’ E L2(0,T ) and u satisfies the boundary conditions.

26

4 The Potential Equation

Write

where S = [0, T ] - [ a , b ] . The integral over S is clearly unchanged by replacing u by 1:. On the other hand, since h ( t ) < 0 we see that the integral over [a, 61 is decreased. Thus J , (0)
.

(6)

This is a contradiction to the fact that u provides the absolute minimum of J , . Hence u 3 0 in [O, T I .

Since

and h ( t ) is an arbitrary negative function, this last fact implies that k ( t , t , ) , the Green’s function, is nonpositive for 0 Q t, t , < T, the desired result. EXERCISES

1. Show that k ( t , t , ) is actually negative for 0 < 1, t , < T.

2. Show by similar argument that k ( t , t , ) has a variation-diminishing property. See R. Bellman, “On Variation-Diminishing Properties of Green’s Functions,” Boll. Un. Mat. I f d . , Vol. 16, 1961, pp. 164-166.

3. Show that one is led to suspect these results by considering the minimiza~ ] d tmay . this be interpreted as a tion of J 3 ( u ) = ~ ~ [ ~ ‘ ~ + ( u - h ) Why smoothing operation?

4. Extend the foregoing argument to cover the equation ~

J

- u - 2u3 = q t ) , J

u(o) = u ( ~ =) 0.

27

6. Two-Dimensional Case

5. What can we say about the solution of the equation U.

- e-u

- h(t),

u(0) = u ( T ) = 0 ,

if h ( t ) 2 O? 6. Show that the same argument establishes a similar result for the difference equation u , + ~- 2u,

+

u,-I

+ q n U n = h,,

n

associated with the minimization of

=

1,2, ..., A'-

1,

UO = UN =

0,

xy:,' [(u,+, - ~ , ) ~ - - q ~ u , ~ + 2 h , u , ] .

6. Two-Dimensional Case The same method of proof can be used in higher dimensions. Consider the two-dimensional case. Let

be the quadratic functional associated with the equation u,,

+

uyy

+ dx,.Y)u

=

h(x,y),

= 0,

(X9.Y)

E

I-.

(2)

We suppose that the quadratic part of D(u) is positive definite. Under this assumption we wish to show that h 3 0 implies that u < 0. We proceed by contradiction. Let u 3 0 in R , E R and consider the new function

As before, we see that D(v) < D(u), a contradiction. Thus h 2 0 implies

u

< 0, whence as before the Green's

function is nonpositive.

EXERCISES

1. Show that the Green's function is actually negative for points inside R .

2. What is the corresponding variation-diminishing property? 3. Extend the result to suitable nonlinear equations as in the one-dimensional case. 4. What is the analogue of the result of Exercise 6 at the end of Section 5?

28

4 The Potential Equation

7. Discretization One powerful approach to the study of the properties o f differential equations is the use o f associated difference equations. This procedure is particularly useful numerically when we contemplate the use of analog or digital computers. However, it is equally valuable for theoretical purposes. The fundamental observation is that the approximation U(X+A,Y)

~xx(X,Y)

+ U(X-A,Y)

- 2U(X,Y)

A2 u(x,Y+A)

>

+ u ( x , Y - ~- ~ u ( x , Y ) Y

A2

U,,(X,Y>

(1)

converts the potential equation into a system of linear algebraic equations. This transformation, of course, is the beginning of the real problem, that o f obtaining useful analytic and computational results from the linear system. 8. Rectangular Region Let us see what this involves for a rectangular region (Fig. 3). Let integers M and N be chosen and the positive quantities A and 6 be determined by M A = a, N 6 = 6. Let urn,,= u ( m A , n 6 ) ,

m

=

0, I , ..., M , n

The original boundary condition, U(X9.Y)

=dX,Y),

(X,Y> E

=

0 , I ,..., N .

r,

(1)

(2)

yields the corresponding boundary conditions UO,,=

UrnN

=

g(0, n6)

9

n

= 0,1,

..., N ,

.q(rnd, h ) .

'obr7pl

We suppose t h a t g ( s , y) is continuous so that g(0, 0) and g ( m A , N 6 ) are unambiguously defined by the foregoing. la.bl

FIGURE 3

I0 01

lO.01

29

10. Associated Minimization Problem

Upon setting x = mA, y relations,

= n6,

the potential equation yields the set of

u((m+ 1) A , n6) + u ( ( m - 1) A , n6) - 2u(mA, n6) A2

Hence we obtain the linear difference equations urn+ 1,n +urn - I ,n -2vrnn

A2 m = 0 , 1 , ..., M ,

+

um,n

+ I + urn, n - I - 2 v m n -

d2

n

=

0,1, ..., N .

- 0,

(5)

These relations, together with (3), constitute a system of linear algebraic equations for the quantities vmn.

9. Rigorous Aspects If A and 6 are both sufficiently small, we suspect that

u(mA,n6) z urn,,.

(1)

This can be established rigorously; references will be found at the end of the chapter. EXERCISE

Show that, under the assumption that u possesses third derivatives, this is a stability result for the solution of (8.5).

10. Associated Minimization Problem It is necessary to consider the existence and uniqueness of the solution of the linear system in (8.5). We can answer these questions readily by noting that the linear difference equations are the variational equations associated with the quadratic form

30

4 The Potential Equation

where rn ranges over 0, I , 2, ..., M and n over 0,1,2, ..., N . The boundary values are determined as in (8.3). Since Q M , N ( uis ) clearly of positivedefinite nature, it possesses a unique minimum value, attained by the unique solution of (8.5).

11. Approximation from Above The foregoing minimization problem, in Section 9, is obtained from the original minimization problem, (associated with D(u)), by restricting attention to functions u for which u, and u,, are constants over the rectangles

rnd<x B

to mean that A - B is positive definite. We will first show that Q is positive definite. Let x be an arbitrary nonzero vector. Then, using the inner product

whence Q is positive definite. We will proceed with the proof inductively. We have AN-1 = I -

[Z+ el-'.

(3)

Since

Q > 0,

(4)

it follows that (5)

Z+Q>Z.

Thus, [ I + Q ] is nonsingular and A,definite, we have

exists. Since [ I + Q ] is positive

[I+

el-' > 0 ,

(6)

[I+

el-' < I .

(7)

0 < AN-1 < I .

(8)

and from ( 5 ) Thus

44

5 Dynamic Programming and Elliptic Equations

Assume now that for some R < N - 1

and therefore

Since we have 0 < AR-1 < I , which completes the induction. Thus, all the matrices we must invert are positive definite and therefore nonsingular. EXERCISE

Show that ( X + Y ) - ' z X - ' - X - ' Y x ' - '

if Y is small.

8. Stability We will take stability to mean that the effect of an error made in one stage of the computation is not propagated into larger errors in latter stages of the computation. In other words, local errors are not magnified by further computation. Let us first examine the matrix recurrence equation A, = I - [I+ Q

+AR+I]-',

(1)

and let us assume that a small error has been made. It follows that we are actually employing the recurrence relation

A", = I -

[I+ Q

where A"R

=

A,

+LR+~]-', + ER,

(2)

(3)

45

8. Stability

is the desired solution plus an error term error at the next stage is given by ER

=

[I+ Q

+

L'i~+l]-'

ER

. Using (1)-(3)

- [I+ Q +AR+1

+

we find the

ER+l]-l.

(4)

Q +AR+~]-'.

(5)

After some manipulation, this expression becomes ER

=

[I+ Q

+

A",+~]-'ER+I[I+

Under the assumptions that the initial error is sufficiently small and that the matrices are not ill-conditioned, we have by (5.5)

[I -

ER

+ 1 1 ER+

1

Cr - A R +

11

*

(6)

Taking norms in (6) we can write

I/ ER 11

/I2 11 E R + l 11

/Iz-

*

(7)

Since all the matrices are symmetric (as a consequence of theory and the choice of numerical procedures) we can use the norm

IIMII = P " ( M 2 ) ,

(8)

where p ( M 2 )denotes the modulus of M Z . Then (7) becomes

11 ER 1

P(Cz-

AR12)

/I E R + l 11.

(9)

However, from Section 6 we know and thus Finally from which we can conclude that the matrix equation is stable. Using the same approach we can analyze the vector recurrence relation bR

= [z-ARl(bR+l

+rR).

(13)

Again, if an error has been made we are actually employing the recurrence relation &R = [ r - A R 1 ( b " R + l +rR), (14) with b " ~= b R e R . (1 5 )

+

46

5 Dynamic Programming and Elliptic Equations

I n accordance with what we did above we can assume that the value of I - A R in (14) is essentially exact. Carrying out the same operations as before we find eR = [IfARI eR+ (16 1 1 9

and upon taking norms

11 " R //


2)

(t,t)

12. Deferred Passage to the Limit We pointed out in Section 8 that essentially all of the errors in the final result are local errors. If we do not choose h too small, something to be avoided on grounds of storage and time, then most of the error is local truncation. Let us assume that the solution of the discretized problem, u(x,y,h), depends in an analytic fashion on h2, and see what use we can make of this, u(x,y,h) = u(x,y,O)

+ u , ( x , y ) h 2+ u 2 ( x , y ) h 4 + ....

(1)

The quantity of interest is u(x,y , 0). One estimate of this is naturally u(x,y,O)

=

(2)

U(XYYY4,

with an error that is O(h2).One way to improve this estimate is to use a smaller value of h. Since, however, the computational time is proportional to 1 /h4, there are obvious disadvantages to this direct procedure. Let us use ( I ) to obtain a better estimate than that appearing in (2). Write u ( x , ~h/2) , =

~ ( X , Y , O )+ u1 ( x , ~ )(h2/4) + u ~ ( x , Y(A4/ ) 16) + ... . (3)

Then u(x, Y , 0 ) =

3 C4u(x, Y , h/2) - u (x, y , h)l

+ 0(h4) .

(4)

Thus we solve the problem twice and calculate (4) at the points common to each grid. Some typical results are given in Table 11. The result computed by series solution is

u(iy$) = 0.4320283.

(5)

50

5 Dynamic Programming and Elliptic Equations

TABLE II

a

~~

0.43 105 0.43178 0.43197 0.43201 (27) 0.43202 (44)

1 16

jt I

64 1 128

0.43202 0.43202 0.43202 0.4320283

These results clearly indicate that the deferred approach to the limit technique yields better results than going to smaller step size, without significantly increasing the required time and storage. It bears repeating that this technique may not be easily applicable to the iterative techniques since, unless we do an excessive number of iterations, the errors may consist of more than local truncation.

13. General Linear Equations Given a region R with boundary

r

the equation

with the boundary conditions

is the Euler equation associated with the minimization of the functional a

n

subject to (2). If we perform the same discretization as in Section 2 duijjax = auij/ay =

+ o(h), C(ui,j+l - uij)/’hl + o ( h )

C(ui+l,j -

uijlihl

3

and define gij = g(ih,jh)h2,

cpij = cp(ih,jh)h2 ,

51

13. General Linear Equations

then the discretized version of (3) is

Now if we define the matrix Q , vectors rR and scalars s, as before and further define the diagonal matrices GR and vectors qRas GR

= diag(gRl,gR2, '..,gR,M-I),

(7)

(PR = [ ( P R j l ,

then the inner product form of (6) is

We can now proceed in the same manner as before. We define

Using the principle of optimality, we obtain the recurrence relation

subject to

Thus we find that the minimizing uR is uR =

[I fQ

+ GR + A R + [I-' (u + b R + 1 + r R +(PR)

while the A, and 6, satisfy the recurrence equations

Y

(14)

52

5 Dynamic Programming and Elliptic Equations

with the initial conditions A, = I ,

b,

(16)

= UN.

We have again discarded the equation for C , since we are only interested in the u,. Thus, we solve ( 1 5 ) with the initial conditions, (1 6), and store the results. Then, we can rewrite (14) as This is an initial value problem starting with u o . The details are left to the reader. I f we go back through the proof of nonsingularity and stability, it should be clear that a sufJicient condition to ensure both the nonsingularity of the matrices A, and the stability of the method is This guarantees the positive definiteness of the quadratic functional. 14. Irregular Regions

We have already noted that if we have already solved a given equation over a particular region and retained the matrices A,, we can solve the equation again with a new set of boundary conditions with very little effort. We can also note from Section 11 that the matrices A, are independent of the forcing function cp(x,y). The matrices A, constitute the Green’s function for the discrete problem and are functions of only the equation, without the forcing function, and the region. Furthermore, it is clear that if we have the matrices A,, A , - , , ..., A , , these are sufficient to solve problems over the truncated regions as shown in Fig. 2. For this region we will need A , , A,..., A K + This argument indicates that in many applications it will be profitable to keep a file of the matrices A , of various dimensions on tape.

M

Kh

FIGURE 2

Nh

Nh

FIGURE 3

Kh

53

14. Irregular Regions

Now we would like to consider a building block technique for irregularly shaped regions. In Chapter 7 we will discuss a direct method for these regions. Suppose we have a region such as shown in Fig. 3. The region can be broken up into two rectangles, I and 11. Let us assume that we are trying to solve the potential equation and we have already had occasion to solve the rectangular problems (I and 11). Ifwe further assume that the rectangular problem has been solved left to right in region I and right to left in region 11, we then have already computed the solution of

where the matrices are of order M - 1. We have also already solved

where these matrices are of order L - 1 . We would like to show that we can use these matrices to compute the solution over the combined region even though the boundary conditions may have changed. For a given set of boundary conditions we can define r R and FR and then solve

R = 192

~ R = [ I - A R ] ( ~ R - I + T R ) ,bo=Uo,

y...)

N-1,

(3)

and

6,

=

[I-

AR]

( 6 ~ ++ 1 FR),

6,

=

R

UK,

=

N

+ 1, ...)K -

1 , (4)

since the orders of the quantities in each equation are consistent and u 0 , u K ,rR and F, are given by the boundary conditions. The difficulty lies at uNwhere the number of grid points changes. However, if we can determine uNthen we can obtain the solution in region I by means of the relation UR

= [I-AR]UR+I

R

+bR,

=

1 , 2 , . . . , N - 1,

(5)

and the solution in region I1 using UR = [ r - A R ] u R - ,

We will now show how We can write u, as

U,

+

6

~

9

R = N + 1 ,..., K - 1 .

(6)

can be determined.

uN =

[I].

(7)

54

5 Dynamic Programming and Elliptic Equations

where

We want to make this distinction since y consist of interior points but w is given by the boundary conditions. Thus we need only determine y. Using the definitions of Section 4, we denote the minimum value of the functional in region I byfN- ( u N )and the minimum value of the functional i n region 11 by gN+ ( y ) . From the quadratic nature of the functional we have* fN-l('!N)

gN+ 1 ( Y )

:])

=.-I([

=

= (AN-luN~uN)-(2bN-l~uN)+cN-l~

(9)

(AN+1 Y , Y ) - (2bN+ 1 Y ) + C N + 1 . 3

If G ( u ) is the discrete functional over the entire region, then using additivity of the functional we see that

This is an application of the idea of the minimum convolution. Since

([

fN-

:I)

and gN+ I ( y ) are in a convenient form we can easily find y .

We partition A N - and b N p 1as follows,

where A , and 6 , have order L - I . Now we can expand (10) and obtain min C ( u ) = min [ ( [ A ,I :u.

1

-

(26N+

1 >

+ A N +~ I Y , Y )+ y ) ] - 2b2 M') + 9

+ A:,] )+',.Y) (261 It', w ) + cN- + c N + .

([A12

(-422

w

-

1

1

Y)

(12)

This expression can be differentiated to find the minimizing y , which is, since A N - 1 is symmetric, y =

C A I 1 + AN + 1 1

( A 1 2 w-

bl

- 6N

+ 1) .

(1 3)

*CR and cR should be defined as in Section 5 . These quantities will not appear in the final expressions for uR and need not be calculated.

55

15. Higher Order Equations

Using this value of y , (5) and (6) are initial value problems starting with uN and y respectively. If then the matrices A, and AR are available, we first calculate 6, and 6,, an easy task. Then we solve for y by (13) and finally solve (5) and (6) for the desired uR. All of the operations require a small number of computations compared with the amount of effort required to solve the irregular region problem from scratch. 15. Higher Order Equations Our technique is not restricted to second order linear equations. Consider, for example, the static deflection of an elastic plate under transverse loading as described by the biharmonic equation uxxxx

+ 2uxxyy+ u y y y y

=P

(1)

7

a fourth order elliptic equation. The boundary conditions for (1) are usually given in terms of conditions at the edges of the plate. For a clamped plate we have U(X,Y) = 0 , (2) at the edges, and a condition on the normal derivative U,(X,Y) = 0 , (3) at the edges. It is well known, and easy to verify, that ( I ) is the Euler-Lagrange equation associated with the minimization of the functional

J ( 4=

1s

[(uxx

+ UYJ2 - 2 P U l d X

(4)

7

R

subject to (2) and (3). We can attempt to find approximate solutions of ( I ) by treating a discretized version of (4).Proceeding as before, we replace u x x , uyy and uXy in (4) by finite difference approximations. In a direct manner we obtain a functional equation of the form fR(',w)

=

min[G(uR> 9' w)+fR+l(uR?w)l 9

UR

(5)

where

u

= uR-1,

w

=

uR-2.

We can then simplify considerably by using the quadratic nature of (4) as before. We leave the details to the reader.

56

5 Dynamic Programming and Elliptic Equations

A simpler approach goes as follows. Let us define a function u by 2' =

u,, + UYY.

(6)

where u is a solution of ( I ) . It is easy to verify that 2) must satisfy the equation u,,

+

Now if we define a vector variable

oyy

=P*

(7)

by

~t'

we find from (6) and (7) that

w,,

+ wyv = A w + s,

where

Appropriate boundary condition for W J can be found from (2) and (3). Since (8) is the Euler equation associated with the minimization of the functional

J(w)

=

jj

[(w,, w,)

+ ( w v ,"J + ( A w , M')

- 2(s, w)] dx d y ,

(9)

R

we can easily solve a discrete version of (9) by the techniques developed in Section 1 1 . The only difference will be that for the fourth order problem, the matrices and vectors will now be of order 2(M- I ) . We will carry out the details of this problem by invariant imbedding in the next chapter. It is easy to show that the method will always provide the solution and that it is computationally stable. EXERCISES

1. Derive the matrix recurrence equation corresponding to (8). 2. Show that the necessary inverses always exist.

3. Derive the associated vector recurrence relation. What is the initial condition for this relation if we are solving the problem defined by (I), (2) and ( 3 ) .

57

16. Distributed Control

4. In many physical situations we are given bending moments and normal forces along a portion of the boundary. Consider for example a rectangular plate with the edge x = 0 subject to bending moments f ( y ) and normal forces g ( y ) while (2) and (3) hold on the remaining edges. Thus, we will have boundary conditions of the form

- u,, (0,Y ) - v u y y (0,Y ) = f ( Y ) > u,,,(o~Y) + ( 2 - ~ ) u x y y ( 0 ~ = Y )g(JJ), on x = 0, where v is the Poisson ratio, a physical constant of the material. Instead of (4) we must consider here the more general functional

+ 2 J gudy + 2 J fu,dy,

in order to consider forces f and g on an edge. Rederive the dynamic programming equations in terms of the variables M ( x , Y ) = - u,, - vuyy,

V(x,

v) = u,, + (2 - v) uxyy.

See the article by N. Distefano listed in the bibliography at the end of the chapter.

16. Distributed Control Problems involving distributed control in general give rise to partial differential equations. Suppose we consider the problem of choosing a control function u = u(x, t ) which minimizes =

u,

u(x,O) = f ( x ) ,

subject to u, = u,,+

lTl'

J(u, u)

[g(x) u2

+

021

dx d f ,

u ( l , t ) = u(0,t) = 0 .

(1) (2)

We suppose that g(x) > 0. We can easily verify that the Euler-Lagrange equations corresponding to (1) and (2) are u, = -v,,+g(x)u,

(3)

u, = u,,+u,

subject to u(x,O) = f ( x ) ,

u(1,t) = u(0,t) = 0,

v(x, T ) = 0 ,

v(1,t) = u ( 0 , t )

=

0.

(4)

58

5 Dynamic Programming and Elliptic Equations

We will use dynamic programming plus discretization to treat problems (1) and (2). The continuous problem can also be attacked directly, but it involves some more sophisticated concepts. Let us define uij = u(ih,j6),

where Mh = 1,

N6

=

T.

We let ui and ci be respectively the M - 1 dimensional vectors j = 1,2 ,..., M - 1 ,

ui = [ U i j ] , Ui

= [Uij].

We can replace ( I ) and (2) by the discrete problem of finding

subject to ui+l

=

[ I - p Q ] ui

+h ~ i ,

where Q is as defined previously, G = diag[g(h), ... ,s(2h),g((M -

1141,

and p =h/6’. The initial condition for (6), u o , is given by (4). If we consider the sequence of variational problems N

Bibliography and Comment

59

We can now derive the minimizing u, namely U =

-h[I- h2A~+1]-'A~+1[Q I -] ,

and a recurrence equation for A , A R = G+[I-pQl[z-[z+h2A~+11-1][z-PQl,

with

A, = G .

We can show that under suitable restrictions on the ration p that the necessary matrix inverses always exist and that the method is stable. In Chapter 10 we will see that by changing our original discretization slightly we can ensure existence and stability for any positive value of p. EXERCISES

1. Show that if we discretize the equation u, = u,,

+u

as

+ [ I + pQ]-'ui,

u i + l= [ I + p Q ] - ' ~ i the recurrence relation for A , becomes

AR = G - k P [ I - [ Z + h 2 P A , + , P ] - ' ] P ,

where P = [Z+pQ]-'.

2. Show that this equation has a solution and is computationally stable for any positive value of p . BIBLIOGRAPHY AND COMMENT

This discretization is standard. It has long been noted that starting with the variational problem will lead to symmetric difference schemes. See Section 2.

J. Todd, ed., Survey of Numerical Analysis, McGraw-Hill, New York, 1962. Sectiun 4.

A number of discrete dynamic programming problems are considered in

R. Bellman, Introduction to Marrix Analysis, 2nd ed., McGraw-Hill, New York, 1970. Section 5. The corresponding results for the semidiscrete, or method of lines, technique appear in

60

5 Dynamic Programming and Elliptic Equations

E . Angel, “Dynamic Programming and Linear Partial Differential Equations,” J . Math. Anal. Appl. Vol. 23, 1968, pp. 628-638.

Scction 7. In most references, the Perron-Frobenius theory of nonnegative matrices is employed to prove existence and stability. See, for example, R. Varga, Matrix Iteratioe Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. Section 8. Ill-conditioning has not been a problem in practice with matrices of order less than 128. Section 10. See

F. W. Dorr, “The Direct Solution of the Discrete Poisson Equation on a Rectangle,” SIAM Reo., Vol. 12, 1970, pp. 248-263. Scction 12. See

L. F. Richardson, “The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with Applications to the Stresses in a Masonry Dam,” Philos. Trans. Roy. Soc. London Ser. A. Vol. 210, 1910, pp. 307-357. Other convergence accelerating techniques can be found in

J. Todd, ed., Survey of Numerical Analysis, McGraw-Hill, New York, 1962. Section 14. See E. Angel, “A Building Block Technique for Elliptic Boundary-Value Problems Over Irregular Regions,” J . Math. Anal. Appl., Vol. 26, 1969, pp. 75-81. This technique is similar to a tabular method suggested in L. V. Kantorovich, V. I . Krylov, and K. Y. Chernin, Tables for the Numerical Solution of Boundary Value Problems of the Theory of Harmonic Functions, Ungar, New York, 1963. Section IS. See N. Distefano, “Dynamic Programming and the Solution of the Biharmonic Equation,” lntc~rnat.J . Nianer. Meth. Engrg., Vol. 3, 1971, pp. 199-213.

Scction 16. See A. P. Sage, Optinnini Sj.stenw Control, Prentice-Hall, Englewood Cliffs, New Jersey,

1968.

R . Ucllman, Introduction to the Mathcrnatiral Theory of Control Processes, It: Nonlincar Processc..~,Academic Press, New York, 1971.

There are many interesting questions connected with the Dirichlet functional D(ir). See R . Bellman and H . Osborn, “Dynamic Programming and the Variation of Green’s Function,” J . Math Mech., Vol. 7, 1958, pp. 81-86, and other papers by Osborn.

Chapter 6

Invariant Imbedding

1. Invariant Imbedding During the past ten years, the theory of invariant imbedding has been used to derive analytic and numerical results in a number of different fields, atmospheric physics, transport theory, wave propagation, to mention a few. From our point of view the invariant imbedding approach will allow us to replace linear two-point boundary value problems which often have associated unstable computational algorithms by initial value problems which possess simple stable algorithms. The method can be applied to cases where there are no associated variational processes. We can give a motivation for the theory of invariant imbedding by considering the following problem. Suppose we consider a neutron transport process in a rod of length L (Fig. I). We will let u ( t ) denote the flux of neutrons to the right at t , and v ( t ) the flux to the left at t . The input fluxes, u(0) and v(L)are presumed given. If we consider the particle interactions between t and t + d where A I , and then let A + O , we can readily derive differential equations 61

62

6 Invariant imbedding

relating u ( t ) and u ( t ) . In many cases, where there is no interaction between particles in the flux, these equations will take the simple form.

+ Bu, = CU + D v ,

u' = Au U'

(1)

v (L) = c2. u(0) = c1, The two-point boundary value problem above is linear, and its solution can be made to depend on the solution of a linear system of algebraic equations. We suspect, however, that usual methods applied in (1) may be computationally unstable unless care is exercised.

FIGURE 1

Suppose we now consider the problem from a different point of view. The flux to the right, u ( t ) , at some point of the rod consists of two components, the part of u(0) which has been itself transmitted and the part of u ( t ) which has been reflected back. If we let T ( t ,L ) and R(t, L ) represent the transmission and reflection coefficients respectively of a rod of length L at t , then u ( t ) = T(t,L)u(O)+ R ( t , L ) u ( L ) . (2) A similar expression holds for u ( t ) . We see that (2) is a direct consequence of the linearity of (1). If we consider how the reflection and transmission functions change as the length of the rod is increased from L to L + d L , we can derive differential equations for R ( t ,L ) and T ( t , L ) as functions of L. These equations will in general be stable initial value problems as far as most numerical methods are concerned, and thus can be easily solved numerically. The method of invariant imbedding has been successfully applied to many classes of problems including ordinary differential, integral, and integro-differential equations. We shall be interested here mostly in difference equations.

2. The Riccati Transformation Consider the coupled linear difference equations, ui+, =

AiUi

+ BiUi + P i ,

u i f l = Ciui+ D i u i + L ,

i

=

1,2,..., N - 1 ,

2. The Riccati Transformation

63

where ui and vi are M-dimensional vectors. We will assume that two-point boundary conditions ug

= C,

UN =

(2)

d,

are specified. We seek a solution to (1) in the form ui = Riui

+ si,

(3)

where the matrix Ri and the vector si are independent of ui and vi. The existence of solutions of this form is a result of the linearity of (1). We will return to this point in the next chapter. We can rewrite (3) as Di+l

= Ri+lui+l +si+l

7

(4)

and use (1) to relate R i + and si+ with Ri and si.Using (1) and (4) we have Ciui

+ Divi + A = R i + l[ A i u i + Bivi + ei] + s i + l,

(5)

and substituting for vi with (3) we find [ C i + D i R i ] u i + j ~ + D i s= i R i + , [ A i + B i R i ] u i + R i + l ( e i +B i s i ) + s i + , .

(6) Since this equation must hold for all ui, we match coefficients in ui to obtain the relations

[Ci+ Di Ri]

+ Bi Ri] , A + Di si = R i + (ei + Bi si)+ si+ , =

Ri+ 1 [ A ,

(7)

or Ri = [ R i + Bi - D i ] - ' [C, - R i + A i l ,

si = [ R i + l B i - D i ] [ j , : - s i + l- R i + l e i ] . Since (3) holds for all ui and vi we have

d = RNuN

+ sN,

(8)

(9)

This relation must hold regardless of the second boundary condition. Thus, we must have RN = 0 ,

sN=

C .

(10)

We now have initial value problems for the Ri and si starting at i = N and going backwards. To obtain an initial value problem for the ui we use (3)

64

6 Invariant Imbedding

in the first equation of (1) to get ui+, = [Ai+BiRi]ui+ei+Bisi.

(1 1)

This is an initial value problem for the determination of ui starting with 240

=c

(12)

and using the values of Ri and si computed from (9) and (10). We will delay any discussion of existence of the necessary matrix inverses and of computational stability until we consider the particular values of A i , B,, Ci and Di which will arise from the discretization of partial differential equations. EXERCISES

1. Start with the representation ui = Riui

+ si

and derive an initial value problem for the determination of the Ri and si.

2. Start with the representation u ( t ) = R ( t ) v ( t ) + s ( t ) and derive initial value problems for the solution of (1.1). 3. Single Sweep Methods We have established the fact that we can replace the coupled difference equations subject to two-point boundary conditions by two sets of initial value problems. Unfortunately, it appears that we have to store the results of the first problem, the determination of the Ri and s i , to solve the second. It turns out, however, that under many circumstances the invariant imbedding approach will yield initial value problems without a n y storage requirement. We will illustrate this by means of a simple example. Consider the scalar difference equations

subject to the boundary conditions 2.40

=

0,

UN

=

1.

(2)

Since ( I ) is linear we can easily consider more general boundary conditions or add a forcing function using superposition. To emphasize that the

65

3. Single Sweep Methods

process described by (1) and (2) has N stages, we rewrite (1) and (2) as

+ biUi(N), u i + l ( N ) = CiUi(N) + d i u i ( N ) ,

U i + , ( N ) = aiui(N)

u,(N)

=

0,

(3)

uN(N) = 1 .

Now suppose we consider the same equation subject to the same boundary conditions except that there are now N + 1 stages, U i . , ( N + 1) Ui.,(N+

=

QiUi(N+ I ) + b , u , ( N + l ) ,

1) = CiUi(N+ 1 ) + d i U i ( N + l ) ,

u , ( N + 1)

=

0,

u M + l ( N + 1) = 1 .

( 4)

As a consequence of linearity, the solutions to ( 3 ) and (4) can differ only by some multiplicative constant k , i.e., U i ( N + l ) = kUi(N),

q ( N + 1)

=

kDi(N).

Setting i = l J in (5) and applying the boundary condition

M N )= 1, we see that k = u , ( N + 1)

or Ui(N+ 1)

=

u,(N+ 1) Ui(N),

ui(N+ 1) = uN(N+ l)ui(N).

Thus we have our fundamental invariant imbedding relation, a description of how the solution changes as the process increases in duration. Let us define r, = U i ( i ) . (9) If we let i = N in (4) we find rN+1

1

= aNUN(N = CNUN(N

Setting i = N in (8), we have u,(N

+ 1) + b,u,(N + 1) + 1) + dNuN(N + 1).

+ 1) = u,(N + l ) r , .

We can now solve (10) and (1 1) for r N + in terms of rN

>

66

6 Invariant Imbedding

Since

(13)

u,(N) = 0 for all N , we have the initial condition for (12) r, = 0 .

(14)

I f W E return to the last section, we see that (12) is a scalar version ofequation (7). However, we have still not used the full power oftheinvariant imbedding method. Suppose we are interested in finding u k ( N ) and u k ( N ) for some specific values of k and N . We solve (10) for u N ( N + 1)

Next we use (1 5) in (8), obtaining

To find u k ( N ) and u r ( N ) we proceed as follows. We solve (12) for N = I , 2, ..., k . Then we adjoin (16) with the initial condition Uk(k)

%(k) =

= rk,

(17)

+

and solve (1 2) and ( I 6) for N = k, k 1, . . ., N - I . Although there has been no storage required, we have obtained only the values u k ( N ) and u k ( N ) . If we were also interested in the values u , ( N ) and v l ( N ) , we would adjoin another set of equations like (16) to (12) and we would start the third set at n = /with the initial conditions

u,(l) = r l ,

u,(l)

=

1.

(18)

Although this procedure is very useful, we will usually assume that we have sufficient storage to carry out our original procedure. However, the reader should keep i n mind that the procedure we just described can be used as an alternative method and may in many instances prove superior.

67

4. Discretization EXERClS E

Derive the equivalent results for the vector equations of Section 2. Hint: Consider the matrix system Ui+ 1 ( N ) = Ai U i ( N )

vi+

1

(N) =

+ Bi V i ( N ) ,

ci U , ( N ) + Di V , ( N ) ,

U,(N)

=

0,

U,(N+ 1 )

=

U,(N)V,(N+ l ) ,

& ( N + 1)

=

&(N)V,(N+ 1).

V,(N)

I,

=

and show that

What physical interpretations do these equations have?

4. Discretization We are now ready to return to the solution of elliptic partial differential equations. Once again as an example of the method we will use the potential equation u,, uyy= 0 (1)

+

taken over the rectangle

(2)

OU r n - l ) ] .

(4)

This important result is another restatement of the principle of optimality. We have succeeded in relating the problem over the region of Fig. 1 1 with that over the region of Fig. 13. We should expect a result of this form since when we go from the region of Fig. 12 to that of Fig. 13, we add two additional terms to our discrete functional and must minimize over one additional interior point. Solving (4) numerically, a topic we shall return to shortly, will enable us to proceed from the region of Fig. 13 to that of Fig. 14 by a sequence of scular minimizations. However, we would also like to find f l k by scalar operations. To accomplish this, we start by definingf,,,(u,, ..., 0,- 1) as fO,k(U1

9

...)

I)

=fm,k+

1 (’l

*

‘”Y

urn- 1)

9

(5)

an obvious choice upon consideration of Fig. 14. Now consider the regions in Fig. 15 and Fig. 16. In both regions the minimizations are carried out over the same interior points. If we let

111

16. Matrix-Vector Equations

FIGURE 12

FIGURE 13

FIGURE 14

denote the minimum of the discrete function over the region of Fig. 15, and using (9,let f O k ( u l , ..., urn- be the minimum over the region of Fig. 16, we have

f i k ( v l , ...,

fl,k(ul

1)

...?

= (OI

To determine fin,,- ( u l , .. ., urn-

FIGURE 15

- uk+

1,0I2 +fO,k(V1

9

...?

1).

(6)

we consider the region of Fig. 17. Since

FIGURE 16

FIGURE 17

there are no interior points we have

an expression involving only the { u i } and boundary values. Hence to solve the potential equation we need only find j i k ( u , , ..., v,) for the proper i and n. We do this by solving the sequence of scalar problems frn,n-1

=fO,n-*,fi,n-2,

f k -2

=f o , n -3 > f l , n -

...Y f m - 1 . n - 2 , 3

?

...,ji'I ,k

9

Ak

*

16. Matrix-Vector Equations

We will now turn to the computational aspects of our new method. Let us denote by u the m - 1 dimensional vector u=[ui],

i = l , ..., m - 1 .

(1)

112

7 Irregular Regions

Once again we can establish inductively that f i k ( u , ..., u r n - , ) = f i k ( v ) is quadratic in u, &(u)

+ C(l,k) .

u) - 2(b"*k', u)

(2) In fact, using the notation of previous sections, we can easily verify that .Am,,-

1

(u)

=

= (A(',k)U,

(u, u) - 2(un, 0)

+ (11+ QlUnr un) - (2rn9 un) + s n . (3)

Since as usual we are only interested in the { u i j } , we will not need the values of c ( ' * ~ and ) we will neglect all scalar terms. From (15.5) we have A(0.k)

= A ( m , k + 1)

9

With no loss of generality we can take the notation A(l,k)= (a$f.k')

,

b(O,k)

= b(m,k+l)

A(*,k)

to be symmetric. Let us use

b ( W

= (b!Lk))

.

(5)

(6)

Then by (1 3.6) we have

Let us write (1 5.4), using ( 2 ) , as

We can differentiate this expression to find the minimizing w ,

u,w = u,+

1

+ by-

rn- I

-

2 + ajf-l,k)

C

i= I if1

*,iui

(9)

113

16. Matrix-Vector Equations

We shall now use this result in (8) to find A(',k)and b"*k).To simplify the notation, let us write

+ p, = (a',u) + p ,

0 - u1 = (a, u ) 0 - UI-1

where by (9) i = 1,

a. =

i=l-1.

otherwise,

(a,v)

"'

0

+B,

i =I

- 1,

otherwise.

Expanding (12) and writing out all terms in u we find (a,

d2= (44

>

(14)

114

7 Irregular Regions

where

A

Likewise

= (aij) = ( a i a j ) .

(a‘, v)2 = (A’%, 0) ,

and

= ( a(i1j-

+ ( a j < - p + 2Ui) aj + a; a;).

1.k)

Finally we determine from symmetric, i.e., A ( @ ) = (a!!.k’) IJ =

I n a similar manner we find P k= ) [by’]

=

(19)

by requiring this matrix to be

+ qp).

+(&f.k)

[ - p(2ai

+ b j 1 - I . k ) + b (1l--11 s k )

+ a;) -

(20)

q ( 1l -, 1 .jk )

~il.

(21)

and vectors b(’vk)in a Thus we can compute and store the matrices straightforward manner. Then we use (9) with o = uk,,- to construct the solution.

17. General Regions It is clear that using the ideas of the last two sections we can treat general two dimensional regions by a sequence of scalar minimizations. Our main difficulty is notational. For instance, for the region of Fig. I8 we might ..., r,,,) to denote the minimum value of the adopt the notation Ljk(v,, discrete functional over this region. Proceding as before we would be led to a functional equation of the form f;jk(ul,

..., v,,,-

=

min [ ( v j - w)’

+ (vj-

-

w)’

W

...)

+ L , j - ~ , ~ ( v 0~j -~2 , W , v j ,

...) v m - 1 ) I

(1)

115

17. General Regions

y;..-I FIGURE 18

R

FIGURE 19

for j > i, and other relations similar to those of the last two sections for values of i, j , and k along the boundaries. However the most interesting and important application of these results is to the solution of partial differential equations over three (or more) dimensional regions. Let us consider, for example, the region of Fig. 19, where

b = Mh, c = Lh. If we let u denote the solution of the potential equation a = Nh,

u,,

+ uyy+ u,,

=

0

9

(2)

(3)

over this region with the value of u specified on the boundary, then we can replace (3) by the variational problem

subject to the same boundary conditions as (3). We now discretize (4) and consider only the values of u at grid points. Let { u j k } denote these grid points, i.e.,

uiJk= u(ih,jh, kh).

(5)

We now apply dynamic programming to the discrete version of (4) which is

(6)

We define N - 1 > M - 1 matrices U, in an analogous fashion to the vectors uR of two-dimensional case, i.e., ul = CuijJ

.

(7)

116

7 Irregular Regions

Applying the principle of optimality we obtain a functional equation of the form

f;(v) = mj:C(Qu~> U ! ) - ( ~ R L ,u R ) +

sL+h+l(uR>l,

(8)

where the inner product of two matrices is defined as

We can now use the fact that once againJ;(V) must be quadratic in V ,

h ( V ) = ( A I K V > - 2 ( 4 , V > + c,.

(10)

Hence we can easily derive matrix recurrence relations for the matrices A and B. Unfortunately the matrices we must invert will be of order ( N - l ) ( M - 1 ) since o u r imbedding shows how the solution changes as a whole surface of grid points is removed as in Fig. 20. Thus for even moderate values of M and N the matrices we must invert can be too large to handle in most computers. We will see in later chapters how this difficulty also arises in the invariant imbedding approach. Suppose now we consider the region of Fig. 21. We can letJ;,(v,, ..., v,) now denote the minimum of the discrete functional over this region where

v.I

= u r. , k + l , l

(1 1)

which are values along the boundary. If we proceed as with the two-dimensional case we can remove the M - 1 points as in Fig. 22 and relate&(v) to,f;, + ( v ) . Now since we are removing M - 1 points at a time we will have matrices of order M - 1 to invert. Finally if we consider Fig. 22 we see we can remove only one mesh point at a time. Thus with this approach we can solve the three-dimensional problem by a sequence of scalar minimizations. It is a tedious but direct exercise to derive recurrence equations for all these approaches.

,

/$a

kiih -

!h”

G

F I G U R E 20

F I G U R E 21

F I G U R E 22

Bibliography and Comment

117

BIBLIOGRAPHY A N D COMMENT

Section 2. For the potential and Poisson equations, various approaches have been suggested for the irregular region problem. See for instance B. L. Buzbee, G. H. Golub, and C . W. Nielson, “On Direct Methods for Solving Poisson’s Equations,” SJAM J. Num. Anal., Vol. 7, 1970,pp. 627-656. Section 3. See

E. Angel, “Discrete Invariant Imbedding and Elliptic Boundary-Value Problems Over Irregular Regions,”J. Math. Anal. Appl. Vol. 23, 1968,pp. 471484. Section 7. The Sturmian separation theorem is discussed in R. Bellman, Zntroduction to Matrix Analysis, 2nd ed. McGraw-Hill, New York, 1970. Section 8. See S. C . Mikhlin and K. L. Smolitskly, Approximate Methods for the Solution of Diffiwntial and Integral Equations, Amer. Elsevier, New York, 1967. Section 9. Further numerical results are contained in E. Angel, “Dynamic Programming and Partial Differential Equations,” Ph.D. Thesis, Univ. of Southern California, 1968. Section I I .

See the list of references in

R. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. Section 12. See E. Angel, Invariant Imbedding and Three Dimensional Potential Problems, Electronic Sciences Laboratory, Univ. of Southern California, USCEE-325, 1969. Section 13. See N. Distefano and J . Schujman, Nutnerical Solution of Boundary- Value Problenis in Structural Mechanics by Reduction to an Initial- Value Formulation, Earthquake Engineering Research Center Rept. EERC 69-4,Univ. of California, Berkeley, 1969.

Section 14. The various relationship between Riccati equations and linearity are explored in

E. D . Denman, Coupled Modes in Plasmas, Elastic Media and Parametric Amplifiers, Amer. Elsevier, New York, 1970. R . Redheffer, “Difference Equations and Functional Equations in Modern Mathematics for the Engineer: Transmission-Line Theory, E. F. Beckenbach, ed., 2nd Ser., McGraw-Hill, New York, 1962. W. T. Reid, “Solution of A Riccati Matrix Differential Equation as Functions of Initial Values,” J. Math. Mech., Vol. 8, 1959,pp. 221-xxx. W.T. Reid, Riccati Differential Equations, Academic Press, New York, (to appear).

118

7 Irregular Regions

A. McNabb and A . Schumitzky, “Factorization of Operators-111: Initial Value Methods for Linear Two-Point Boundary Value Problems,” J . Math. Anal. Appl.,

Vol. 31, 1970, pp. 391-406.

Section 15. See

E. Angel and R. Bellman, “Dynamic Programming and Reduction of Dimensionality for the Potential Equation,” J . Math. Anal. Appl. (to appear). Section 17. Another approach to the irregular region problem is to regard an irregular region as imbedded in a regular region. Constraints are used to fix the boundary conditions on the boundary of the irregular region which is now inside the chosen regular

region. The advantage of this approach is that methods which work well for regular regions can be extended to very general problems. See

E. Angcl, frregrtlur Rc>gionsarid Constrained Optitnization, Electronic Sciences Laboratory, Univ. of Southern California, USCEE 71-27, 1971. B. L. Buzbee, F. W. Dorr, J. A. George and G. H . Golub, The Direct Solution of the Discrete Poisson Equution 011 Irrc,gular Regions, Computer Sciences Department, Stanford Univ. STAN-CS-71-195, 1970.

Chapter 8

Special Computational Methods

1. Direct versus Iterative Methods Our two basic approaches, dynamic programming and invariant imbedding, are both direct (i.e., noniterative) methods. As a consequence of these approaches we were required to invert sequences of symmetric matrices of general form. Hence, for a square region our methods require 0 ( N 4 ) multiplications and divisions. We argued that the relatively high number of operations was compensated for by a number of factors such as the ability to solve similar problems with almost no additional computation, the ease with which these methods can handle irregular regions, and the freedom from the choice of critical parameters. On the other hand, the iterative finite difference techniques require only 0 ( N 2 ) operations per iteration and totally avoid matrix inversions. These are certainly important advantages in their turn. In this chapter we will derive a number of iterative methods directly from dynamic programming and invariant imbedding. Although we will show how these methods are equivalent to the standard methods, the approach we use will allow us to extend the iterative techniques to irregular region problems. 119

120

8 Special Computational Methods

We will also discuss some modifications of our direct methods which will apply to constant coefficient equations. With these modifications we will be able to avoid matrix inversions. Our example will again be Laplace’s equation on the square. We use this example so that we can conveniently demonstrate the technique used t o analyze iterative methods. We will make extensive use of the characteristic values of the matrix Q , using Kronecker products to aid the analysis. Although these techniques are restricted to constant coefficient equations, we can justify their use on two accounts. First, it is through a thorough understanding of the constant coellicient equations that we will be able to approach more difficult problems. Second, these equations are important in their own right, as evidenced by thc extensive body of literature on the potential, Poisson, a n d biharmonic equations.

2. The Characteristic Values of Q Before we proceed further, we must dispense with a few mathematical preliminaries. We have seen that the analysis of all our methods depended on the positive definiteness of the matrix Q. At this point, then, it will be useful to examine the characteristic values of Q. Let Q be of order N . We write Q as (1)

Q = 2I-P, thus defining the matrix P as

Thus, if we find the characteristic values of P we will have the characteristic values of Q. Let the matrix F, F =

(3)

(Lj) 9

be defined by

Jij

=

sin [ i p x / ( N

+ I)]

sin [ j q n / ( N

+ I)],

(4)

where p and q are arbitrary integers. Then, if

PF

= (aij),

(5)

2. The Characteristic Values of Q

121

we have by direct multiplication aij = {sin[(i+ 1) p n / ( N + I)]

+ sin [(i-

I ) p n / ( N + l)]}

x sin [jqrr/(N+ l)] =

2sin[izrp/(N+ l)] cos[pn/(N+ l)] sin [jqn/(N+ l)] .

(6)

Let D denote the diagonal matrix

Then we find, setting FD = (bij),

(8)

bij = sin [inp/(N + I)] sin [jnq/(N + l)] u j .

(9)

that Thus, PF

=

provided that uj

=

(10)

FD,

2cos[jpn/(N

+ l)] .

(1 1)

Since p is arbitrary, the characteristic values of P are given by i = 1,2,..., N .

ui = 2cos[in/(N+ I)],

(12)

and thus, the characteristic values, p j , of Q are the quantities pj = 2{1 - C O S [ ~ X / ( Nl)]}, +

j = 1,2,..., N .

(13)

3. Kronecker Product In the previous chapter we briefly mentioned the Kronecker product. In this chapter we will use these products extensively. Let A be an Ndimensional matrix and B an M-dimensional matrix. The Kronecker product A 0 B is defined as the NM-dimensional matrix C

=

A @B

=

(aijB).

(1)

122

8 Special Computational Methods

We can easily show that the Kronecker product thus defined has a number of convenient algebraic properties. For instance

0( B 0 C ) = ( A 0 B ) 0 C , (2) ( A + B ) @ ( C + 0 ) = A 0 C + A 0 D + B @ C + B @ D , (3) A

(A@B)(C@D)=(AC)@(BD).

(4)

EXERCISE

Prove (2), (3), and (4). 4. Kronecker Sums

Let us consider the matrix C defined by

C = I @ A + B @ I.

(1)

If A has order N and Border M we will use the correct orders of the identity matrices i n ( I ) , so that C has order M N . We will assume that A and B are both real symmetric since this is the only case we will use later. Let { g i } and {Pj} be the characteristic values of A and B respectively. We write A and B as

A

=

TAT’,

B

=

SBS’,

(2)

where the prime denotes matrix transposition and

and TT’ = I ,

SS’ = I .

(4)

We now use (2) i n (1) and apply the properties of the Kronecker product noted i n Section 3,

c = 10 (T2T’)+ SBS’) 0I

+ ( S B S ’ ) 0(TT’) = ( S 0r)( I 0A) (s’0 T ’ ) + ( S 0T ) ( B 0I ) (S’ 8 T’) = ( S 0 T ) ( I 0 A + B 0I ) ( S 0 T ) ‘ . =

( S S ’ ) 0 (TAT’)

(5)

123

5. An Example

We can directly verify that

( S @ T ) ( S @ T)'

=

I

and I@A+B@Z is diagonal. Thus, the characteristic values of C are the diagonal entries of Z @ A + B @ I , the MN numbers ai+Pj. We will use this result extensively. EXERCISE

Show that the block tridiagonal matrix

A C

can be written in the form

D

B

C

A

B

1, u=[iyly1.

= [ I @ A + L @ C + U @B],

where L and U are defined as

L=

B A

[;;*;;.

0 1

0

0

1

.:..

5. An Example

Consider our discretized version of the potential equation ui+

- [4Z - P ] ui+ ui- - ri = 0 ,

(1)

where we have replaced Q by 21-P as in Section 2. We will assume that the region is a square. Thus ui is of order N , uo and u N + *are known. We saw in Chapter 6 that (1) could be written as

124

8 Special Computational Methods

a relation of the form

A M= r ,

(3)

where these quantities are of order N 2 . Closer investigation shows that the matrix A has the structure A = - I @ P- P

0 I + 41 0 I .

(4)

Thus we can write the solution to (1) as

If we write P as

P

=

FDF,

where

then we see that ( 5 ) becomes u = [F

0 F ] [ - I 0 D - D 0 1 + 41 0 I ] - ’ [F’ 0 F ’ ] r .

(8)

However the matrix [ - I @ D - D @ 1 + 4 1 @ 1 ] - ’ is a diagonal matrix whose elements are the numbers 1/(4-vi--vj). Since we already have shown that pi = 2cos[in/(N

+ I)]

(9)

and F = {sin [in/(N + l)] sin[&/(N

+ l)]} ,

(10)

(8) gives a direct method of solving the potential equation on rectangular regions. In the literature, this technique is known as the “tensor product” met hod. EXERCISE

Show that the tensor product method requires 0 ( N 3 ) operations.

125

6. Another Direct Method

6. Another Direct Method Suppose we are solving either the potential equation or the Poisson equation on a rectangular region. We saw i n Chapter 6 that the invariant imbedding approach led to the following equations for the vector uN

..., R , i = 0, 1, ..., so

i

R i + , = [2Z+ Q - R i ] - ' , si+l =

Ri+l(si + Ti),

= O,l,

= 0, =

c,

(1)

and ui+t =

i Pi+t

=

Ri+l ui7

n,n + 1, ..., N ,

=Pi

+ UiSi, U,

p, = 0 ,

=

R,.

Finally we obtained the desired solution in the form

(3)

UNd+PN.

U, =

If we write Q as (4)

Q = FDF,

where D is diagonal and F is the symmetric orthogonal matrix we introduced in Section 2, then we see inductively that (5)

Ri = F R i F and where Ri and

ui= F U , F , U iare diagonal. Si =

Let us define vectors Si and Fs~,

pi =

pi by

Fpi .

(7)

Now it is easy to verify that these new quantities satisfy the equations

and

=

[D-Ri]-l,

Si+l =

Ri(Si + &ri),

R,

= 0,

S , = Fc,

i

=

0 , l ) ...,

i

=

0 , 1,...,

(8)

-

ui+l= R i i i i , pi+l ii,

-

=

pi + U i S i ,

=

R,,

p,

i = n,n = 0.

+ 1, ...,

(9)

126

8 Special Computational Methods

Finally we obtained the desired solution by

Since all the matrices except F are diagonal, we have eliminated all the matrix inversions, and since D and F are known we can easily carry out these computations. EXERCISE

Show that this method requires O ( N 2 )operations for the potential equation and 0 ( N 3 ) for Poisson's equation.

7. Diagonal Decomposition

I n Chapter 5 we discretized the variational problem corresponding to the potential equation in the following fashion

and we found the quantities uR by applying dynamic programming to (I). Using the principle of optimality we obtained the functional equation for the truncated minimization problem, fR

("1

=

min [( Q U R un

3

uR)

- (2rR

9

UR)

+ SR + (uR

- u, uR

- v>

+fR + 1 ( u R ) l

9

(2) with u =

UR-1.

(3)

The function,f, ( u o ) provides the minimum of ( I ) . It is clear by observation by (2) that the matrix (2 provides the only interaction between the components of the state vector, u R . If we were to go through the development of Chapter 5 again, we would see that it is precisely this interaction which forces us to carry out a sequence of matrix inversions. Suppose we have an approximation to the solution, that is, a set of vectors {u;')}. We can write Q as Q = 21- P ,

(4)

127

7. Diagonal Decomposition

where P is again the matrix

P = ,

Pij

=

1,

li-jl = 1 ,

(5)

0 , otherwise.

Note that our approximation has the property that

an important property which we would not have if we used the simpler approximation (PUR

3

(10)

(PUkO’, u R ) .

UR)

We now use (8) in (2), and we get the approximate formula fR

(u) = min [ ( 2 U R , uR)- ( ~ P u ~ O ’ ,uR) UR

+(UR-u,UR-2))+fR+1(UR)1

+ ( P U ~ ” , uL”)

- (2rR,uR)

+ sR (1 1)

9

or fR(u)

=

min [ ( 2 U R

UR)

9

UR

- (2FR

Y

UR)

+ :R + (uR

- u? U R

- U,

+f R +

1 (UR)l

9

(12) where FR

= rR

+ PUP’,

?R

= SR

+ (PUkO’, uko’)

f

(13)

The functionfR(v) defined by (12) is clearly quadratic in u. Thus if we set ~ R ( u )= ( A R U , U ) - ( ~ ~ R , ~ ) + ~ R ,

(14)

we find, by the same procedure we used in Chapter 5, that the minimizing uR is given by UR

=

[31+

AR+l]-’(v

+ + FR

bR+1)

3

(15)

128

8 Special Computational Methods

and that the functions A , and h, satisfy

with h,

A, = I ,

(17)

= u N + ~ .

I t is clear inductively that ,4, as defined by (16) is diagonal. Thus no matrix inversions are necessary. In fact to get A , from A R + I only N scalar divisions are necessary and the solution of ( 1 5)-( 17) requires only 2 N 2 multiplications and divisions. The vectors {u,} given by the above procedure will o f course only be a next approximation and we must iterate on (15) and (16). Using ( 1 3 ) we can rewrite our equations i n the form A,

0'"+" K

-

I - [31+ A R + l ] - l ,

=

[f - A , ] (hg::)

+ r~ + Pug'),

A,

I, bjyk")

(18)

=

= %+I?

(19)

and p + l ) R+ I

=

[ I - A,]u',+"

+

@+I)*

(20)

Since the equation for A , is independent of the boundary conditions and the approximation, we need only solve it once. We can show that the method defined by ( 1 8 ) - ( 2 0 ) converges. However we will delay the proof until we have developed a number of iterative methods from the invariant imbedding approach. It will be easier to prove convergence from this approach.

8. Point Iterative Methods The discretization of the potential equation led us to the matrix vector difference equation uR+l -

2uR

+ UR-1

- QuR

+rR = 0 ,

(1)

where u0 and uN are known. Suppose we now use the fact that Q can be written as

Now we can write ( I ) as

129

8. Point Iterative Methods

which immediately suggests the simple iterative scheme Uk"'

1)

=

t [Pug' + uk"!, + u f l , +

TR],

(4)

where {uko)}are initial guesses. Note that although (4) retains our matrix vector notation, each component of up' is computed independently from each other component. Thus, all our computations are scalar computations. The procedure in (4) is known as the "point Jacobi method." If (4) converges, it will converge to the solution of ( 3 ) . Let the error, ek")be defined by e (Rk )

= UR-Up.

(5)

We immediately find that e$" satisfies the difference equation e R( k +') = $[Pe',k)

+ e$! + e$! J ,

(6)

with e(k) 0

= (k)

eN+l

=O,

(7)

for all k . Let us define the N dimensional vector e(k)as e(k)

=

[eIk)].

(8)

Using Kronecker products, we can now write (6) as e ( k + l )= $[P 8 I + I @ P ] e ( k ) .

(9)

If we let T denote the matrix T

=

t [ P @ I + I @ P] ,

(10)

so that

if and only if Fortunately, we have shown that the characteristic values of P are the numbers

vi = 2cos[i7r/(N

+ l)] .

(14)

130

8 Special Computational Methods

Hence we find that

p ( T ) = cos[71/(N

+ l)].

(15)

Thus, the method will always converge. However, for N large

cos[n/(N

+ I)]

E

1 -f[n/(N

+ 1)12.

(16)

This means that for fine discretizations the convergence will be extremely slow. The point Jacobi method requires that we wait until the end of the kth iteration before we use the new values we have just computed. If we assume that we always iterate in the order of increasing x and y and that we use new estimates as soon as possible, we get the Gauss-Seidel method,

=I

u g + l )=

$[LU$+1)+

uuk"'+ p;' + 42 + J,RI 1

(17)

9

where L and U are defined by

L

0 1

0

0

1

0

0

.

.... 1

0

.. ..

u=

0

1 0 1

1 0

0

Equivalently, L and U are a strictly lower triangular matrix and strictly upper triangular matrix such that (19)

L+U=P. EXERCISES

1 . Show that the significant matrix T for the Gauss-Seidel given by

T=[I@(41-L)-L@I]-'[I@

method is

U+U@I].

2. Show that p ( T ) = {cos[n/(N+I)]}'.

9. The Successive Overrelaxation Method We saw that for both the Jacobi and Gauss-Seidel methods, the moduli of the significant matrices will be close to one if the grid spacing was small. T o get around this problem, we can use an acceleration parameter.

131

9. The Successive Overrelaxation Method

Let

fig")

denote the value yielded by a Gauss-Seidel iteration,

cg+1)= ' [ ~ ~ g + +l )uu$)+ ug':)

+

+rR1-

U(k)

R+l

(1)

Then we will obtain the new iterate, ug"), by taking a weighted average of this value and the last value, =

U (Rk + ' )

(j$jg+') + (1 - o)ug',

(2)

where o is our acceleration parameter. Eliminating fig'') from (1) and (2) we find that the point relaxation method is defined by [41-

OL] u f + ' ) - o u g ? : )

=

[4(/ - o )+~ O U ] u g ) + oug,

+ orR. (3)

The reader can easily verify that the significant matrix is T

=

[ I @ (41- wL) - OL @ I]-'[I@(4(1 - o)Z+

oU)

+ OU @ I]. (4)

The characteristic values of T are the solutions of the determinantal equation

IT-ilZl = 0. (5) In the exercises we show how this equation may be solved. The result is that the ith characteristic value of T, l i ,must satisfy the relation l i

+

0

- 1 = pioli"/2,

(6)

where again pi is the ith characteristic value of P. Note that for w = 1, we find the characteristic values of the Gauss-Seidel method. We must now find for what value of w the modulus of T is minimal, that is we must find

p ( T ) = min maxl liI . m

i

(7)

Considering the complex mapping defined by (6), Young has shown that the optimal o is determined by 0 2 v 2 = 4(0 - l),

(8)

where v is the largest characteristic value of P . The method thus defined has been called "successive overrelaxation" since the optimal o satisfies the relation 1 u , + l + ( / O Q +QOI>u,l.

=

(6) Solving for zi,+ u1+ 1 = [ I

1,

we find

+ ( r / 2 ) ( I OQ + Q 0 I ) ] - ’ [ I -

( r / 2 ) ( I OQ

+ Q 0 01

Thus, the significant characteristic values are the numbers

~

1

.

(7)

where p i is a characteristic value of Q. Once again since 0 < pi < 4 , we have

O

(7)

~~~

* W e assume that the appropriate boundary conditions are used in the solution of

(1) and (2).

7. Alternating-Direction Implicit Methods

155

and

Since Q is tridiagonal, each of these systems can be solved in a simple manner. As we saw in Chapter 8, we will have scalar versions of the matrixvector computations necessary to solve elliptic equations. We can analyze the stability of this method by considering the homogeneous version of (7) and (8). Combining the two equations we have

It is clear that the matrices Z-(r/2) Q and Z+(r/2) Q commute. If we let T denote the matrix T

=

[ I - (r/2)Ql[I + (r/2)PI- ',

(10)

then from (9) we have

u,+,= T'u,T'.

(1 1)

Since the characteristic values of Tare the numbers

where p i is a characteristic value of Q, we have 0


= f(x,

(1)

over a region of very general type. Our aim then is to solve an equation such as (1) as the limit of the solutions of a sequence of linear equations of the foregoing form. This involves the method of successive approximations, the general factotum of analysis. The simplest variant of the method is that of Picard. In place of (1) we consider the sequence of linear equations ( a + 1)

uxx

+

1)

= exp(u(~)),

(2)

where d o )is specified, the initial guess. Similarly, in place of (1.2) we consider the sequence of linear equations $+l)

= u,,( " + I )

+ u;+ + g(u'"'), l)

(3)

again with do)specified. Presumably, this resolves the matter. There are, however, two major sources of difficulty. In the first place the convergence is slow, geometric convergence at best. I n the second place, unless u(O) and d o )are carefully chosen there is no convergence at all. For these reasons we will employ a different variant of successive approximations, one generated by considerations of the theory of dynamic programming.

3. Quasilinearization A fundamental class of equations in the theory of dynamic programming has the form maxT(u,u) U

=

0,

164

11 Nonlinear Equations and Quasilinearization

where T is linear in u. The function u represents the return function, while u represents the policy function. In solving ( I ) we can employ approximation i n function space, the classical approach, or approximation in policy space, a new approach, with a number of desirable features. If we guess a policy, u,, the equation for the associated initial return function u, is linear,

T(u,, uo)

=

(2)

0.

The origin of (2) suggests a method of successive approximations. Let u , , the new policy, be determined by the equation

W,, u d = max U u , , 4, u

(3)

and let u , , the new return, be determined by the equation

W ,, v1)

=

0,

(4)

and so on. I n many important cases we can demonstrate that the convergence is monotonic, and quadratic. This will be the case below. 4. An Example

Consider the equation u2-220. Let us employ the identity u2 = ( u

+ (u -

From this it follows that u2 3 v’

for all u, and that

0))’

=

v2

+ 2u(u - v) + ( u - v)’.

+ 2v(u - v) = 2uv - v 2 ,

u2 = max[2uu - 21’1, D

with the maximum furnished by u = u. Hence the nonlinear equation takes the form max[2uu - u2 - 21 li

From this it follows that 2uv-u2-220,

=

O

165

5. The Equation

for all c. If we are interested in the positive solution, we have u

< (2 + 2)/211,

(7)

whence u = min[(v2 a>O

+ 2)/2u].

(8)

If we employ (4) as a basis for a method of successive approximatior., and proceed as above we are led to the celebrated recurrence relation

un+1

=

+

(9)

(un2 2)/2u,.

This coincides with that obtained from the Newton-Raphson method, which assures us of quadratic convergence. EXERCISES

1. Let g(u) be convex. Show that g(u) = max[g(v)+(u-v)g’(v)]. a

2. What is the geometric interpretation of this analytic result? 3. What is the multidimensional version of the result of Exercise 1 ?

4. Consider the two relations 0

= f(u) = f(un)

0

=

+f ’ ( u n )

(u - an)

+ 0E(u - un>’I

9

f ( 4 +f’(un) ( u n + 1 - 4

3

and thus deduce quadratic convergence to the root off(u) 5. The Equation u,,

= 0.

+ uy,, = uz

Let us now consider the numerical solution of the equation u,,

+ uyy= u 2 ,

(1)

=f,

(2)

over a region R , where u

on r, the boundary of R. We will focus initially on this simple example since it will enable us to demonstrate conveniently the convergence properties of quasilinearization. As we have noted, the most common method for establishing the existence of a solution of (1) and (2) is Picard iteration. We replace (1) by the

166

11 Nonlinear Equations and Quasilinearization

where u(i+l)

=f,

(4) on r. Since [u'~)]' is a forcing function for (3), we can examine the convergence of (3) via the Green's function representation of

Although we will not carry out the details, it can be shown that (3) does converge to (1) provided that do)is a sufficiently good initial approximation. However, the convergence is geometric. By this we mean that if we define a functional norm

llsll = maxIg(x,Y)I? R

(6)

then

// u - & + I ) //

= O(ll u

-d

i y )

.

(7)

Thus Picard iteration does not in general lead to an efficient numerical method. Let us now consider the quasilinear approximation to (1). We are led to the iterative method (i+l)+ u(i+l) = 2u(i+l)u(i) YY

UXX

-

[u(i)]2

9

(8)

with (9)

on r. 6. A Differential Inequality We wish to relate solutions of the differential inequality u,,

+ uyy+ qu 2 0,

subject to .l4

=f,

167

7. Monotonicity

on the boundary, to the solution of the equation

v,

+ vyy+ qv = 0,

(3)

=f,

(4)

subject to

v

on the boundary. We will assume q is such that the quadratic functional H(u) =

1

/uxz

+ u:

- qu2] dx dy

,

(5)

R

is positive definite. A simple sufficient condition for this is q < 0. For any function u which satisfies (1) and (2) we can write u,,

+ uyy+ qu - p

= 0,

where

p 2 0.

(7)

Then if we define w as

w w must satisfy w,,

=v-u,

+ wyy + qw + p

= 0,

(9)

where w = 0,

(10)

on the boundary. Then since p is nonnegative and the associated Green's function is nonpositive (a result established in Section 6 of Chapter I), we have w 2 0 ,

7. Monotonicity

(11)

168

11 Nonlinear Equations and Quasilinearization

on r, do)- f , converges monotonically, for i > 1, to the solution of u,,

+ uyy = u 2 ,

(3)

=f,

(4)

with 24

on

r, provided that this solution exists. First we will prove

We have already indicated that at any point

Hence

u,,

+ uyy 2 2UU"'

- [U(')l2

.

(7)

Let us define a sequence of functions w ( ~ + 'by ) w(i+l)

Then by (l)-(4),

it,('+ I )

=

- u(i+l)e

satisfies

and thus u 2

di).

Wc will now establish a similar relation between u ( ; + ' ) and d i ) .Taking u = d i +' ) in (3) and (6) we have (;+I) uxx

+ u y( ;y+ I ) > 2u(i+1)u(;)- cu('l + l ) ] Z ,

(13)

with =f ,

(14)

169

9. Quadratic Convergence

on

r. We now repeat the previous argument to find &+ 1)

2

.(i).

Combining (1 2) and (1 5) we have u

3

&+I)

2 )i(.

3

... 2

u(1)

or monotonic convergence. 8. Maximum Domain of Convergence What is quite interesting about the foregoing is that we have convergence for any choice of the initial function provided that the equation has a solution. We thus have a maximum domain of convergence for the method of successive approximations we have employed. Note again that in this case quasilinearization yields the Newton-Raphson-Kantorovich approximation method, together with monotonicity.

9. Quadratic Convergence As far as numerical computation is concerned, the most important property of quasilinearization is quadratic convergence. We will show that for the problem we are discussing

11 u - u ( ~ + 'I/) < k 11 u - u(') /I

(1)

9

for some constant k . The iterative equation can be written as (i+ 1 )

ux,

+ u,,( i + 1 )

=

[1~(i)]+ 2 2cu(i+1)

- u(i)]

u(i)

9

(2)

with u(i+l)

=

on r. Since u2

=

[u"']2

f,

+ 2u"'[u - u ( i ) ] +

the original equation u,,

(3)

Cu

- u(i)]2

7

+ uyy = u2 ,

(4)

(5)

can be written u,,

+ uyy = [u(')]2 + 2u""u

- u(9]

+ [u - u(i)]2.

(6)

170

11 Nonlinear Equations and Quasilinearization

Subtracting (2) from (6) and defining ,,,(i)

by

M'(~)

- u(i)

=

It is easy to show now that d i + l ) ( x , y )< maxIw'(x,y)I R

(7)

9

11

Ik(x,y,a,b)Idadb.

(1 1)

R

The above result can now be written as, using (7),

jj u - U ( i f 1 ) tl

kl

It u - 24( i ) /I2 ,

(12)

where k , is a constant.

10. Computational Aspects The method of quasilinearization replaces the nonlinear elliptic equation uxx

+ uyy

=

d4

+

(u(i+l)

(1)

7

by the sequence of linear problems uy;l)

+

L12Y+1)=

g(u(i))

- u(i))gu(u(0),

(2)

where (2) is subject to the same boundary conditions as ( I ) . The nonlinear parabolic equation ut =

uxx

+ uyy+ d4

(3)

is treated i n exactly the same manner, i t . , replaced by the sequence u;i+l)

=

UX ( ;X+

I ) + U YY ( i + l ) +g(u(i))

+ ( u ( i + l ) - u('))gu(u(i)).

(4)

For an arbitrary function g(u), not necessarily convex or concave, we might not have the favorable convergence properties we have demonstrated

171

11. Example

for our sample problem. However, for well behaved functions g(u), quasilinearization will converge if the initial guess, d o ) ,is sufficiently close to the solution. In most physical problems it is usually not a difficult problem to obtain a reasonably good initial guess. The problem usually is that of obtaining a reasonable degree of accuracy. We must store the previous solution, di),in order to generate the succeeding solution di+'). Unless the grid size is made exceeding small, the amount of storage required for this is not significant.

11. Example

As a second numerical example let us consider the solution of u,,

on the rectangle 0 < x < 3, 0 < y

+ u,,,

= e" ,

< $, subject to

(1)

(2)

u=o,

on the boundary. The quasilinearized version of (1) is (i+ 1) + &+ , 1) = exp(u(0) LUG+ 1) + (1 - u ( i ) ) ] . uxx

(3)

The initial guess used was 0.

(4) The method converged to four significant places as is shown in Table I u(0) =

for two typical points. The problem was solved again, this time with the boundary conditions (5)

u = 10,

and with the initial guess

u(O) = 6.

Four iterations were sufficient for convergence to four digits. TABLE I

0 1 2 3

0.0 - 0.00707060

- 0.00707072 - 0.00707072

0.0 -0.0060304 -0.00603050 -0.00603050

172

11 Nonlinear Equations and Quasilinearization

This problem was previously solved using quasilinearization and successive overrelaxation. The optimal relaxation parameter was found to be different for (3) and (4) from the optimal relaxation factor for (3) and (5). Choosing a parameter suboptimally can cause the number of iterations to double. Furthermore, the optimal parameter was “optimal” only in the sense that it was kept constant for the entire problem. Certainly this choice ) changing is not optimal for any given iteration since the function c P ( ~keeps From stage to stage. Thus, i t is clear that iterative methods for solving linear elliptic equations are not particularly well suited for use with quasilinearization. However, the direct methods of dynamic programming and invariant imbedding seem to fit in perfectly with quasilinearization.

12. Identification Problems Thus far we have been concerned with only one problem. Given a partial differential equation and a set of boundary conditions, we are required to obtain a numerical solution. Suppose, however, we are interested in some physical process i n which we know that the variable of interest satisfies an equation of the form

+ uyy

u,

= g(u,

4

9

(1)

where u is an unknown parameter or vector of parameters. We are allowed to observe the physical process and to make measurements of u at various points. From our measurements we are required to determine the parameter a so that the solution of (1) agrees with the observed data. I n other variations of this inverse, or identification, problem we are required to identify initial or boundary conditions from observations. Inverse problems arise in such diverse settings as the study of drug distributions in the body, cardiology, and weather prediction.

13. The Least-Squares Criterion We will concentrate on the following problem. We are given a physical process, assumed to be described by the equation 4,

+ uyy+&a)

=

0,

(1)

where a,

a

=

@I,

a,, ..’, 4,

(2)

14. Newton-Raphson-Kantorovich

Method

173

is an unknown vector parameter. We will assume that we know that on the boundary of the region of interest. Suppose we make a set of L observations of the process. Each observation consists of an observed value of u, uf ; the points of observation are ( x f, yf). As a measure of the deviation between the solution of ( I ) and (2) for a given a and the observations, we will use the least squares criterion L

s = I1 Cu(x,,.J+) - u J 2 . = 1

(4)

The identification problem now becomes one of minimizing S over all possible choices of a. Although there are many other ways of choosing a goodness of fit criterion, the favorable analytic properties of the least squares criterion make it a popular choice. 14. Newton-Raphson-Kantorovich Method

It is clear that if the boundary conditions and the function g are sufficiently smooth, the function S is a continuous function of a. To emphasize this point, we rewrite the problem as min S ( a ) = min LI

(I

2 [ u ( x , , y,, a) L

/=I

-

uf]”

(1)

Differentiating the function in ( I ) with respect to a, we find that necessary conditions for a minimum are

(2) or using the gradient

V S ( a ) = 0.

(3)

Our method will be to solve ( 2 ) [or (3)] by the Newton-Raphson-Kantorovich method. Let us define the vector functionf(a) by

f ( 4 = VWa),

(4)

where we have

/(a)

=

[f;.(a)].

i = 1,2,..., k .

(5)

174

11 Nonlinear Equations and Quasilinearization

We start by writing the Taylor series expansion off(a) about a vector b, .f,(a> = fl ( h ) + (0,- 6,) I!?([

+ ... +

(Uk

(b)/dUIl

- bk) [dkf(b)/dU,]

+ (a2 - 6 2 ) C?fI(b)/~~21

+ ...,

(6)

JZ(4 =@ ;! I + ( a , - 61) Caf2(waaII + Jh(4 = L ( b ) + (0,- b,) Cdfl (b)/aa,l + ". .

...?

Keeping only the linear terms, we have the approximation

=

+ J ( b ) (0- 6)

(7)

Cdh(b)/a~jl.

(8)

where J is the Jacobian matrix J

(6) =

Since we are interested i n a root off(a),

.f(4 = 0

(9)

9

our approximation, (7), yields u = b - J(b)-'f(b).

(10)

Since this value of a will be an approximation to the actual root, we use

( 1 0) iteratively,

1)

.

= u ( i ) - J - (u(i)).f(u(i))

(1 1)

We expect quadratic convergence, that is

11

- &+')

Sincef(u) is given as

11

= O(lla -

&y).

(12)

f ( 4= vs(4

the Jacobian becomes the Hessian matrix

J(4=

( . sa , L ? J ) .

(14)

Thus each iteration of ( I 1) requires the evaluation of and Sa, for i = I , ..., k . We will now discuss how these functions are evaluated.

15. The Sensitivity Equations Since we must evaluate Starting with

we will write out these functions in detail.

175

16. Quasilinearization

we differentiate to find

and sa8aJ

=

2

L

C Cua,(x,, Y J U ~ , ( XY~J , I= 1

(

~

(

~

YJ

1

7

- ~ t ) ~ a , a , ( x I YT J I

*

(3)

Thus, we have introduced the sensitivity functions u,, and u ~ , which ~, must be evaluated at each observation point. We will now show that the sensitivity functions satisfy elliptic equations of a special nature. We start with the original equation u,,

+ uyy+ g(u, a) = 0 .

(4)

Differentiation with respect to a, yields (ua,)xx

+ ( u a j y y + gu(u, a) ua, + g a , (u, 4 = 0

7

(5)

Differentiating again we have (ua,a,)xx

+ (ua,a,)yy + gu(u, a) uataJ + g u u ( u , a) u a , ua, + gua, (u,a) ua, + ga,u(u,a) uo, + go,,, (u, a) = 0 .

(6)

Since the boundary conditions on u are fixed, we have u., = 0 9

(7)

and ua,a,

=09

(8)

on the boundaries. We note that these L2+L sensitivity equations are linear and are all of the form

u,,

+ uyy+s,(u,a)u + 4 = 0 ,

(9)

where only the function q changes in different equations. We will exploit this similarity in a moment.

16. Quasilinearization In general the original equation u,,

+ u y y + g(u, 4 = 0

176

11 Nonlinear Equations and Quasilinearization

is nonlinear. Thus, we employ quasilinearization to obtain a numerical solution. We replace ( I ) by the sequence of linear equations ( i + 1 ) + u(i+ y y 1 ) + gu(u(i), a)

uxx

[u'"

- u'"]

+ g(u'i', a) = 0 .

(2)

This equation, as are the sensitivity equations, is also of the form u,,

+ uyy + g u ( u , u ) u + p = 0 .

(3)

Thus, as the sequence defined by ( 2 ) converges, ( 2 ) and all the sensitivity equations differ only in their forcing functions and boundary conditions. This is a tremendous computational advantage. I n Chapters 5 and 6 we saw that the numerical solution of an equation of the form ( 3 ) involved the solution of matrix and vector recurrence relations of the form

starting with the initial conditions where F, is derived from g,,,rR is derived from the boundary conditions and p R is derived from p . Thus, it is clear that since the A, are independent of both the forcing function, p , and the boundary conditions, the matrices A , are the same for the last quasilinear iteration and for all the L2 + L sensitivity equations. The solution of the matrix recurrence equation

-,

is thus computed once, stored, and used to generate solutions of all the sensitivity equations. Since for each sensitivity equation we now need only to compute the b , by (4) and the solution via we have eliminated most of the computation required.

17. Example Let us consider an example of the general problem just considered. Given the nonlinear equation

u,,

+ uyy+ aebu= 0 ,

(1)

177

17. Example

where u =g,

on the boundary. We would like to find a choice of a and b such that S(a, 6) =

c L

I= 1

CU(Xf,YJ

-

%I2

9

(3)

is minimum. First we replace (1) by the sequence of linear problems, for any fixed a and 6, ux ("'x

+

uz'l)

+ a exp(bu("))+ ab(u("+')- ti(")) exp(bu(")) = 0 ,

(4)

subject to =

99

(5)

on the boundary. The optimal a and b are given by the solution of

sa=o,

sb=o,

(6)

Then using the Newton-Raphson-Kantorovich method we generate the sequences a'") and 6'") by means of

where

178

11 Nonlinear Equations and Quasilinearization

The necessary sensitivity equations are found by differentiation of ( I ) to be

+ (uJYy+ ebu+ abebuu, = 0 , + + mebU+ abebuUb= 0

(u,,),, (Ub),,

(ub)yy

( 1 1)

9

and (u,,),, (Ubb)xx

(u,b)xx

+

(ubb)yy

+ (u,,),,~+ 2bebuu,+ ahebuu,, + ab2ebuua2= 0 ,

+ 2aebuub+ 2abueuub

+ au2ebu+ abebUubb+ ab2ebuub2= 0 ,

(12)

+ (uab)yy= UebU+ bebU + aebuU, + abuebUua+ abebUuab+ ab2ebuu,ub= 0 , Ub

uba

with u, = ub = u,, =

Ubb

= u,b =

=

uab

3

0

(13)

on the boundary. Note that all the equations of (1 1 ) and (12) are of the form u,,

+ uyy + abebUu+ p

=

0.

(14)

The procedure is as follows. For a given a = a(") and h = b'") we solve ( I ) via the quasilinear equation ( 4 ) . As the solution, d n f l ) ( x y, , a''"), 6'")) converges, we save the matrices A , , from the final iteration of (4). The sensitivity equations of (1 I ) and (12) are solved using these matrices and the last value of u ( " + ' ) as u to generate the necessary functions. Then (9) and (10) are evaluated and (7) is solved to give 1) =

-

Sbb(a'"), 6'"') S,(U'"),

6'"')

- S,b(a'"),

6'"')

Sb(U'"),

6'"')

S,,(a("), 6'")) Sh,,(dm), 6'")) - [Sab(dm), b'"))l2

b ( " + l ) = b'"' -

9

(15) LyOb

(a'"), 6'")) S, (a'"), h'"') - s b b (a'"), 6'"') S(a'"), b'"') So,(a'"), 6'"') S,,, (a"'), 6'"') - [Sub(a'"), b'"))]

The procedure is repeated until the sequences for a'") and h'") converge. At each iteration we use the last solution as an initial guess, i.e., u(0)(x,y,

I ) ,h ( m +

1))

= uG)(x,y,

b(m)),

(16)

where ii refers to the final iteration of the previous quasi-linear iteration.

179

Bibliography and Comment

This problem was solved numerically on a unit square using (152) = 225 interior points. The results are shown in Table 11. The entire computation took 8 seconds of computer time. TABLE II

Iteration

a(k)

b(k)

0 1 2 3

1.5000 1.6819 1.6805 1.6806 1.6806

1 SOW 2.2737 2.2686 2.2687 2.2687

4

MISCELLANEOUS EXERCISES

1. Introduce the stationary operation, S, defined in the following fashion, S , ( f ( u ) ) =J(u), where u is the assumed unique solution off’(u) = 0. Show that if.f‘(u) is never zero, then

f(u) = s u

+ (u - u ) f ’ W l ’

When is the operation one of maximization and when of minimization? 2. Show that if, in addition,f’(u) is never zero, then we can write r =

su [ u -f(4I f ’(u)l

as a representation for r , the assumed unique solution off(u)

= 0.

BIBLIOGRAPHY AND COMMENT

Section 1. See R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary- Value Problems, Amer. Elsevier, New York, 1965. Section I I .

See

R. Bellman, M. Juncosa, and R. Kalaba, “Some Numerical Experiments Using Newton’s Method for Nonlinear Parabolic and Elliptic Boundary-Value Problems,” Comm. A C M , Vol. 4, 1961, pp. 187-191. Section 6. See

E. F. Beckenbach and R. Bellman, InequaMies, Springer-Verlag, Berlin and New York, 1961.

180

11 Nonlinear Equations and Quasilinearization

Section 12. See J. Casti, D. Detchmendy, H. Kagiwada, and R. Kalaba, “Estimating the Parameters of an Inhomogeneous Medium by Probing with Rays,” in Cumputational Approaches in Applic4 Mechanics, Arner. SOC. of Mech. Eng., New York, 1969, pp. 200-209. E. Angel, “Inverse Boundary-Value Problems for Elliptic Equations,” J . Math. Anal. Appl., Vol. 30, 1970, pp. 86-98.

Appendix

Computer Programs

This appendix contains four sample computer programs and the output from these programs. Although constant coefficient equations are solved, we make no use of the simplifications of Chapter 8. Thus these programs can be easily generalized to include a large class of equations. The following subroutines are necessary : M V ( A ,N , B, C) computes product of square matrix A and N-dimensional vector B and puts result in C. MM(A,N, B, C) computes matrix product A B and puts result in C. Matrices are square and of order N . MIN V ( A , N, D,L, M)-IBM scientific subpackage Gauss-Jordan-matrix A of order N is inverted in place. Determinant is put in D. L and M are N-dimensional work vectors.

181

182

Appendix * Computer Programs

Program 1. Dynamic Programming

Dynamic programming is used to solve the potential equation

u,,

+ uyy= 0

over a square region. The matrices Z - A , and vectors b,, as described in Chapter 5, are generated and stored on a high speed disc. The significant equations are AR=Z-[Z+Q-AR+1]-', bR

= [IARI

(bR+l

A N = Z ,

+ rR),

and UR=

[I-AR]~~R-~+~R.

bN

= uN,

UOOL 0002

0003

0004 00U>

0006 0007 OOOd 000')

0010 001 1 0012

0013 0014

OUlZ 0016

0011

0011) 001')

0020

0021 0022

0013 0024

0025 0026

0027

UUZd

00.2'4 0030

0031

0032 0033

D

x

U

n

E-

D

U U a

n

E'

8 3

U

CT

UL3U,

1

0.0 0.0

0.3 0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0

0.0 0 .O

0.0

0.0042Y O.Ub570 u .0064L 0.12 728 U.UI120

O.lbl>M

0.0155U 0.22670 0.01619

O.Lbla0

0.3201M 0.2 8670 0.02140 0.30 152 0. O L l U l 0.30644 0 . 02 140 0.30152

O.OLOlM 0.26670 O.OAd19 0.26180 0.01550 0.22670 0.U1220

0.0 0.0

0.0

-.

%

0.0

0 .o

0.1812d 0 .00M42 0.12728 0 .00429 0.06510 U.0

0.0

0 .0 0.0

c.0 13876 0.08346 0.01717 0.16057 0 -02488 0.22704 0.U3Lb0 0.L80Y 1 0.03708 0.32188 0.04113 0.35042 0.04361 0.36720 0.04445 0.37272 0.04361 0.36 7L 0 0 -04113 0.35042 0.03 70M 0.32169 0.03160 0.28092 0.02988 0.22704 0.01717 0.16057 0.00876 0.08346 0.0 0.0

J.0

O.d

U.0 U.01339 0 . 10757 O-OLbbL

0.20450 3.03dJb 0.26539 U.04.494 0.34d04 0.0574U 0.3Y440 U.objb3 0.42590 O.Ob747 0.44412 U .Ubd 76 0.45007 0.0b747 0.44412 0.06365 0.42591 0.057w

0.3 Y 4 4 1 U.046Y5 U . 34M05 0.0385b d . 28509

0.OLbbL 0.20450 0.0133Y J.10751 J.O

0.0

0 .d

0.0 U16’90 0.14231 0.03712 d.L647b u.0537a 3 . 3b07 I J.

0.06822 0.43177 J.0799~ 0.4Ml7Y ’ 0.0d85d 0.51468 0.09386 0.53329 0 -09564 0. J 3 ‘ 3 3 2 O.OY38b

0.53330 0.08858 0.51468 0 -0 7 Y Y 5 0.48179 0.06812

0.43178 0.05378 0.36077 0.03 I 1 3 Q.Lb477 d.0169M 0.14231 0.0 0.0

0.0

d. 0

J.U2>17 0.1 Y O Y I U.UZYL4 0.3514Y 0.U7120 0.46144 d. 09OLU 0.3304Y 0.10556 U.3bbLY O.llbd7 0.61772 0.12370 0.63>07 0 . 1260 I 0.64061 0.1237b 0.63507 0.1lbd7 0.61773 U.lU556

0.5 8630 3.09020 0.53650 3.U71LU 0 -46144 0. U4Y L 4 0.35149 0. 0 1 5 1 I

0.1Y69.d 0.0 0. J

0.0 0.0

3.0

0. J

0 .OjL40 0.1938b u.00343 0.4nzaj 0 .09 15 7 0.59701 0.ll5d3 0.66647 0.13531 0.70917 U 14J>O 0.734Mb

0.04124 U.4Y570 0.0d042 3. odMY>

0.15823 0.74d64 0. IbLI4 0.7519Y 0.15823 0. 7 4 8 6 4 0.14956 0.73486 0.13531 0.7L191d 0.11560 0.66647 0.09157 0.5’1 7 0 1 0.06343 0.48283 0.0324b 0-29jMb 0 .O 0. 0

I.

0.1+612

0.82j21 0.11586 0.77729 0.08045 0.68890 0.04124 0.49570 0 -0 0.0

O.U>LU5 lJ1)UUU

0.13121 1.00000

O.lr531

0.11380

U.7 1719 U.14bl2 0.dLjLl U . 1702Y 0.d4Y01 0.16704 O.db389 0.13d40 0.8 7 1 6 ) 0.2OZI)l 0.87437 0.19846 0.87165 0.13784 0.8636Y 0.17030 0.84907

0.0

1.ouoo0

0.182Z I

1.00030 0.21191

1 .00000 0.23305

1.00001)

0.24516

I . OUOOU

r).L+99Y

1.00000 0.24576

I . 00000

d.23305

1.00000 I

0.L1192

.ooooo

0.18251 1 .00000 0.145jA

1 .00000

0.10127 1.00000 0.05205

1.00000 0.0

0.0

186

Appendix * Computer Programs

Program 2. Riccati Transformation Once again the potential equation is solved. I n this case only rapid access storage is used. Thus, as is to be expected, this program executes faster than the previous one. The significant equations are =

[21+

= AR(bR

and U R + ~= A R u R

Q - A R + I ] - ' ,

+ TR),

+ bR.

AN-1 bN-I

=o, = uN,

0001

0002 0003 0004 0005 0006

D

0007

U U a

0009

E'

0008 0010 0011 00 12

0013

0014 0015 0016

001 7 0018 0019 0020 0021 0022 0023 0024

0025 0026

n

8 3

U

CT

002 7 0028 0029 0030 003 1

0032

0033

0034 0035 0036

D

U U a

n

E'

8 3

U

CT

0037 00 38 0039 0040 0041 0042 0043 0044

I. 0U000 0 .u

0.3

0.0

I .I)WJkJU I. 0 0 0 0 0

U.*Y 5 7 0 0.87105 0.29 3 8 0 0.74d64 0. I Y 6 9 1

0.6350~ 0.0 0.0

0.0 0.0 0.0

0.0

0.3 0.0

0.14231 U.533L9 0.1075 I 0.4441 1 0.0b546 U.56719 U. 0 6 3 I U 0.30152 0.U5205 0.24575 0.04124 0 . 1 Y 846 0.05246 0. I 5 b L 3 0.UL517 0. 1L376

0.0

0.0189d O.b'i580

0.0 0.0 0.0

0.0

0.0135Y 0.0674 7 U.0Ud76 0.04361 0.0042Y 0.02140 0.0 0.0

1 .00000 1.00000 0 .OL( 3'16 0. b 6 3 d 9

0.48263 0.13483 0.3514d 0.61772 0.20476 0.51401 0.20449 0.42 5Y0 0.lOU37 u - 3 5 042 0.1272d 0.28070 0.10127 0.23305 0.0d045 0.187d4 0.0034j 0.14Y50 0.04Y24 0.11687 0.05715 0. C d 8 5 8 U .J2662 0.06365 0.01117 0.04113 0.00842 0.0201 8 0 .0 0. Ir

1 .UUlJ30 1.03U00 J.77729 U.64901 U. 5Y 7 J U J.?i)917 U.40143

0.5d6LY U.jbU70

0.4617b 0 .L 8 5 06 J.5944U U . 2 L 104 O.3cldd U.IOl>Lf

0.~6180 U.1453~

0.LlIYI

d.ll5dO u.17030 u. UY I 2 7 0.13331 U.07lLJ U.10558 0.053 l o 0.U7YY5 0.03050 U.U5740 U.UL'tr)Lf

0.03 73d 0.0122U 0.0ldlY 0 .U 0.0

I. 000UU 1 .0J0JJ 3.623.21 3L 1 J. bob4 7

u.dL

d.bbO41

3.55649 0.53049 3.43171 0.43117 J.34804

3.34d04 3 . LdOYL 0. 2 3 3 9 1 0. L L 0 7 0 0.22670 J.IdL51

U.Id25I

3.14612 0.14012 J.lISd0 0.11280 U.UYULU

0.04023 U.0082L 0. d6d.2~' 0.06894 0.04694 3.UjlOO 0.U316J 0.3153U 0.01550 0 .o 0. 0

1.U0UUU 1.0UdUJ J. d 4 9 J 7 u.717~9 3. 7 0 9 1 I 0.291Vl J.5doL9 0.4014c J.4dllo U. 3 6 U I O 3.594-J U - 2 b5Ud 0.5Lldb U.22704 3.Loldu

0.lol5d J.21131

0.14551 J. I I029 U.11500

0.15>31

U.UYl57

0.105>6

0.U7120 J.Ul995 0.3331d J. 0 3 7 4 0

1. UU030 l.JU0UU

0. d o 3 d 9

0. u d d Y o 0.7346J 0.48263 0.6117L 0.3514Y 0.51407 3.20470 J..tL>YJ

0.20450 0. 3 5 0 4 2 0. 10057 0.26070 J. I 2 7 L d U.L53U> 0.11)121 0.1b784 0.3dU45 U.l4Y>o 0.06345 0.11687 0. 3 4 9 2 4 J.OdM5d

0.03712 u. 0 0 3

3

0.33850

3.3206~

0.U5700 0.02486 0.UI81Y 0.J12LU u. 0 0.0

0.041 J.01717

3

U.020 U. 0 0 8 4 L

J

O.U

3.3

1. 0JJJ0 1.0JUOJ

I. 0UJUJ

I . 0UUOU

I) . t l 7 * U

U . O l l O >

3.4YhlU U. 7 4 d b 4

3.L93do U.032JO 0.1Y6jl J. 3 5 3 2 9

U .u

5.14231

0.U

U.4+411 0*107>1 U.3671Y 0.08346

u.72299 U. 6 4 J b U

U .J

U.55931

0.4>UL)O

0.0 0.31.21L 0.U

0.3Ul5L 0.0051~ 0.i4373

I

O.J

0

.u

0.50644 LJ.24999

0.0>103

0.0

0.1Yd40 0.341L4 0.13dLj

0.0

0.05L40

0.0

U. 12370 3.02317

J .0

0.LOLOl 0.16113 0.1LbU7

0.09504 0. U 95do 3.0 0.JloYo 0.UOdlb 0.067~1 0.01359 U. 0 0.04442 0:U430l 0.0 J.OUd7o 0.02181 U.UL143 0.J04.29 0 .0 0. U 0.0 0.0 0 .0

190

Appendix * Computer Programs

Program 3. Invariant Imbedding The potential equation is again solved numerically. In this example the solution i n the middle of the region is obtained using the same grid spacing employed in the previous two examples. Note that the one sweep nature of the computation has eliminated the storage requirement. Thus we solve Ri

= [21+

Si

=

Q -Ri-l]-l,

R, = 0 ,

Ri[si-,+ ri],

so = d ,

and at the desired k we adjoin Ui+1 = UiRi,

uk

pi+l = p i + uisi,

Pk = 0.

Finally we compute the product uk = u r J c + p , .

= Rk,

0001

0002

0003

0004

0005 0006

0007

0008

000Y 0010 0011 0012 0013

0014

0015 0016 0017

0018

0019

0020 0021

0022 0023 0024 0025 0026 002 7 002 8

0029 0030 003 1

b

P

v m

P X

6 3

v

4

L

L

LdMPJTk

L

0032 0033 0034 0335 003b 0037

13 3 u 7 7 dL(1

UL(I UL t H LALL

D

U U

0038 0039 0040 0041 0042 0043 0044 0045

UU46 0047 0040 ou49

0050 0051 U05L

0053 0054

a

E X

bOLUTIUN OF P U T t N T I A L EQUATION 6 Y I N V A R I A N T I H L ~ E D D I N G 1 5 PCIlNTr, I N X I J I R E C T I G N 15 P O I N T b I N Y u I R t C T l t i N

.

dJUNDARY VALJtb UT , U t l r U L , UR 1.00000 1 0uuu0 1 .ou30u 1 .00000

0.0

0.0 0.0

0.0

0. 0

0.0

0 .0

1.03003 0

.u

0.0

1 .UI)OUV

1.0uu03

1.03030

0 .o

0.0

0.0

u. u

u.0

0.0

0.0

0. 0 0.0

U .3

J. 0

0. i)

0.0 i) .d

I. 33uu0

1.00000

0.0

4.0

0 .U 0.0

1.0u00u 1.00000

0.0

0.0

0.0 0. 0

0.0

0.0

1 .0000u

0.0

0.0

0.0

0.02 181 0.30644

1.00000

0 -0

U.0

0.0

iOLUTION AT X =

1.00300

a n

ii’

0. 0 0.0

0.0 0.U

u.u

0.0

8

0.04445 0.37272

0 .1)6a76

3.45007

0.09>04 0.1LbU7 3.~3Y32 0.64061

0.16114 0. L J L 0 1 0. 75299 0. 37407

i)

.L4Y Y

D

U U

‘1

194

Appendix

*

Computer Programs

Program 4. Quasilinearization

The Riccati transformation is combined with quasilinearization to solve the nonlinear equation u,,

+ uyy = u 2 .

Thus the program solves the sequence of linear equations ( i + 1) + u ( i + 1) YY

UXX

- 2u(i)[u(i+1)]

=

- [u(i)]2.

The initial guess u(o)

0

is used. Only the first three iterations are shown since succeeding iterations show no changes in the first five significant digits.

0001

0002

0003

0004

0005

0006

0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019

0020 0021

0022 0023

0024

0025

0026 0027 0028 0029

0030

D

x

U

n -. X

003 1 0032 0 i) 33

00114 0035 003h 0037 0036 003Y

0040 004 1 0042 0043 0044 0045 0 0 46 0047 0048 0049 0050

0054 0055 0056 005 7

0058 005Y

0060

ITERATION 0.0 0.0 0 .o

0.0 0.0 0.0 0.0

0

0.0 0.0

0.u 0.0

0. U

0.0 0.0

0.0 0.0 0.0

u. U

0.0

0.0 0.0

0.0 0.0

0.0

c

0 .0

O.u 0. 0

0 .u 0.0

0.

C

0.0

O.U 0.0

0.0

3.0

0.0 0 .o 0 .0 0.0 0.0 0.0

0.0

O.u 0 .0 0.0

u.0 C.O

0

0.0

u.O J .0

u.u u.O u.0

u.u U. 0 u.0

3.0

0.0

0.0

s

.

0.0

0.0

3.0

J.U

u.O

J.u

0.0

0.0

0 .a

0.1) 0.0

u .u

0.0

0.u

u.O

0.0

3.0

U.3 0. 0

O.U

u. u

O.J

0.3

3.3

0.u 0.0

3.0 0.0

0.0 0.0 0.

0.0

3.3

u

u.0 0.0 0.

u

0.0

0.0

0.3

0.0 0.0 0. d 0.

U

0.0 U.3 U .ll

0.0

J.J

0.J J .U

0.0

1. 0000u

0 .0 1. 00u00 0.0 1 .00000 0.0 1 .uu000 0.0 1 .0u00o 0.1) I . 00000 0.0

u. u

1 .u0000

0.0 0 .o

0.0

1.00000

3. 3

0.3

0.0

0.0

0.0

0 .3

J.U d.O

u

0. 0 0. U 0.0 0. 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3. 0

0.0

0.0 0.0

0.

0.U

0.0

0.0

0.0

0.0 0.0

U.O

0.0

0.0

0.0 0.0

0.0 0.0

0.0

0 .I)

0.0

3.0

3.0

O.U

0.0

U.O

0. i) 0.0 0.3 3.0

3.0 0. 0

0.0

3.0

0.3 3.0

0.0 0.0

3.0

0.0

0.0

0.1)

0.0

2.0

0.0

0

0 .U

u.0

0.0

0.0

u. u

0.0

0.0

0.0

0 .0

O.U

0.0

0.0

U.9

U.O

0.0

0.0

0.0 0.0

3.0

0.0

0.0 0.0

0.0

0 .3

0.0

0.0

0.0

0.0

0.0

0.0 0.0

u.0

0.0

0.3

0.0 0.0

0.0

u.

0.0

0.0

0.0

J. U

0.0

0.0

0 .0

0.0

0.0

u.0

0.3 0 .0

0 .I)

0.0

0.0

u .J

0 .0

0.0

0.0 0.0

0.0

0.0

0.0

0.0

0.0

0 .0 0.0

0.

0.0

0.0 0.0

0.0

0.0

0 .0 0. 0 0.0 0.0 0.0

0.0 0.0

0.0

0.0 1.00000 0 .0 1. u000ll O.u 1.00000 0.0 I . 0u0uo 0.0 1 .0000u 3.0

I.

uou0u

.

u.0 1 000UJ 0 .0

0.

u

-.

Q

OD

1

IT€HATION

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0

0.0 0. 0

0. 3042 J 0.06570 0.00842 O.lz7zd 0.01223 0.18158 0.01530 0.22670

0.01t)lY 0.2b1d0 0.02Ul8 0.28070 0 .OL 1 4 0

0.50152

0.0

0.0 0.0

0.0

0.0 0.0

0.3 U. 0

0 .0Lld 1 0.30b4.t 0.02140 0.30122 0.0201 d 0.28670 0.0ldlY U.L61d0 c.01353 0.22670 O.OlLL0 0.18158 J. U 0 d 4 L 0.1L7Zd 0.00429 0.06570

ti .0

0.0

0.0

0 .0

3.0

0.0

J.O

0.0

0.0

0.0

0.0

0.0

0.03d7b 0.08346 0.01717 C.lbO57 0.024dd 0.22704

U.Ulj>Y 0.10757 0 .0L062 0.2 0420

0.0ldYd 0.1+231 0.03715 0.Lb47o

0.JL2lI 0.19691

3-0j~Ca 0. LY 386 J.06543 0.4d2dA 0.0Y157 0.59701 0 .I 1 5 6 3 0.66647 0.13531 0. 7 0 9 1 7 0. 1 4 3 5 0 3.73480 U. 1 5 8 2 3 0. ( 4 d O . t 0 . 1 6 1 1u 0.7329Y

0.04124 0.49570 0.04043 0.bdMY5

0.03160 0.28092 0.03708 0.321tld O.04113 0.35042

0 -0430 1

0.36720 0.04445 0.37272 0.04361 0.36720 0.04113 0.35042 0.03708 0.32 1 8 Y 0.0316U

0.28092

0.02-dM 0.22704 0.01 71 7 0.16057 O.0JdTo 0.0b34o 0.0 0.0

3.1)3850

0.28 5UY 0.u4d94 U. 3 4 d 0 4 J.U>743 u. 3 9 4 4 0 0.0b305 0.4239U 0 -06 7 4 7 0.444 1 L 0.J6d7b 0.42007 0.06 7 4 I 0.4441 2

0.06303 0.42591 J.U574lJ J.394’vL

0.J4d93 0.34dd> u.03k330 0.Lti5JY J.OLOb2

0.2U45U 0.0ljSY 0.10137 3.0 3.0

0.02378 0.36U77 0.OOd22 0.43177 0.07993 0.4dL79 O.Odd5d 0. 5 1 4 0 d

0.04924 0.3214Y J.U712U 0.46144 J.OY020 0.5364Y 0.1053d 0.58629

0.1 1687 0.61172

O.lL370

3 -09380

0.53329 0.095o.t 0.53932 0.09j86 3.53330 3.0tld58 0. 3 1 4 6 8 3.0749 3 3.4d179 3.06bL2 3 . 4 3 1 78 3.03376 0. 3007 1 J.03715 L3.26477 0.018Yd

0.03207

0.14L31

0.1909L

3.J

0.J

0.126Jl

0.64061

0.123/6

0.15dL3

U.03307 3 . 116d 7 0.61773 0.10536 U.>do30 0. 0 9 0 2 0 OS>3b5U U.UILL0 0.40144 u. U4Y L 4

0.74d6.t 0.14950 0.73480 0.13531 0.70Jla ‘J 115 d 3 0.66641 0 .OY 1 5 / 0.597UL 0.Uo343

0 . 5 Z 14‘4

3.40283

J.JL217

3.0

0.0

.

u.

03L+O

0.29300 0.0 J. 0

0.0

0.0

3.0

0.0

0.0

0.

1 l>do

0.7772’3 0. l 4 b l l

O.dL32L 0.17023 0.d4907 u.ld7dC O.tl63d9 0.1944b 0.d7105 0.20201 0.87407 0. I 9 8 r 6 0.87105 0.187~34 0.d63dY 0.1 / O M 0.84907 0.14olL 0.82321 O.ll>do J. 7 7 7 2 9 0.08042 0.08890 0.041L’t 3.49570 0.0 0.0

0. U 5 L U 5 1 .00000 0.13127 I . oouoo 0.14531

1. 0 0 0 u 0

O.ldL51

1 .J0000

0.211Y1 1 .0000u

0.23505 1.00000 O.LCSI6

1.00000 0 .24Y 99 1.00ou0 0.24370

1 .oooou 0.23305 1 .00000 O.LLIYL 1.000i)O

O.ldL51

1.~0Uti0

0.14331 OJOOJ 0. l U l L 7 1. JOUUU J.U>LdJ 1. 000UJ

I.

O.U

0. 0

ITERATldN 0.0

0.0

0 .0 0.0 0.0 O.J O.U

0.0 0.0 0.0

0.0 0.0

0. 0 0.0

0.0 0.0

0.0

2

0 .u

u.u

(1. J u u l d 0.06451 b.UudlJ (1.11451 b.Ulldb 0.17 7>u 0.Ul5UO

0.~214L 0.Ui 768 U.25531 0.01961 0.27Yb4 u.02019 0.2Y3YY 0.UL113 O.LYd74 o.JLur9 0.LY3YY

0.01561

0.2 1 9 0 4 0.01 ro8 0.15521 0 . U1500 0.22 14.2 0.0lliJb 0.1115U 0.OUd 3 0.12451 0.004 d 0.06431 0.0 0.0

0.3

L.J

0.0

0 .oo d 7 2

0.3~187

3.3ioro

0.15741 3.JL.tlj 0. 2 2 2 3 U u .03 u 72 0.274do J.03603

0.31460 0.Uj39b 0.34254 (1.04L51 0.35851 0.d431d U.56391 0.04237 0.35857 0.03Y ) b 0.34254 O.lJ36J3 0.31466 0.05012 0.27461 0.02419 0.22250 0-01b70 0.15741

O.COd>L 0 -08 167

0.u

0.0

2.0 0.Ul3LL LJ.lU>dl U.UL>JY J.20397 J.U3lV9 3.279d7 J.U4 1 2 0 0.5413U J.J5279 J.5ob5b 3. Ubld-

0.41

bY3

U.LIb>>> u.43.155 U.06079 0.4403u U.0b332 0.45455 0.0bld4 0.41 b 9 5 U.U>57Y 0.58639 0.0473b 0.34130 0.0374Y 0.27967 0.u25n9 0.20JYl

0.015LL U.105dl 0.0 0.0

>.

3.0

3.0 J

d.Ule41

J. 1 4 3 4 2

d.ujbl> J.LoOY7 0.02231 J. J > 3 1 > J.Jo033 0.4245.t J.U?111 u., 1 5 2 1 O.dd609

3.u

1. J

0.J

11.J>lO-I

3.LjLUb J .Ub 1 O d..tl921

u.

J.JOY

291

J.LJ>JOb

I . 3uuuIJ J

10

lo

0.11L7L J.63980

0.ljlo3 J. 7J13U U.

IS>+"

U.5J51L 0. 39ALU

u. I L b 4 3

0.52513 U.OYL')L 0.52843 J.OY120 0.32313 J. 0660 3

0.739 17 U.12601 0.143Y7 U. l > 3 d u 3 . I5Y77 0.14340 0.72044 0.13103 J. l U l 5 U 0.11271 3.6>960 u. 0 O Y l d 0.2Y 1 7 0 0. 0 b l M J 0. 4 792L J.U5104 0. L Y L U O 0.0 0.0

3.20512 d.07111

3.4 73.22 3.0bb33 0.4L454 3.05131

3.33512 3.0561 5 J.26097 0.01847 0.14042 0.0 0.0

I.

J. 1 > 3 a a

u. U.

I

OYL!YU

11JULJJ

14lCII

.JUJUU

u.

I ldJU

1 .J)oUuu

3. 2 J U b 3

1.030JU U.2LIUj

1. u)oouJ

J. 25 9 5 3 1. U J U J d u. L . t 3 4 3 1 .000UJ J .LA Y33 1. 000uLJ 3.22104 1.3UOUU 0.2J653 1 .u30u3 0.1 I d U U 1.OUUOU U. 1416.2 1. 0 J 0 0 U 0 . UYUYU 1. u 0 0 0 u O.U5U6b 1 .Ji)030 lJ. 0 0.0

h)

8 I T r h A T I [JN

3 0.1,

0.0

U.J

0.0

0.0

0.3041d ir.iib431

0.0 0.3 0.0

0.0 0.0 0.0

0.0 0.0

0.0 0.0 O.J

0.0

0.0 0.0

0.0

0.00d1

I)

0.12451 0.U11ao 0.17750 0.01 53 b 0.22142 0.01 r b d 0.25551 0.01361 O.27Yb4 d.ULO79 U. 2Y3YY O.ULIl9 0.29d74 U.OLd7J 0.2Y3YY

0.01Y61

0.2 7Yb4 0.01 I b l l 0.25551 0 .U 1306 0.22142

0.01186

0.17750 0.U0dld 0.12451 0 .OO418 0.0643 I 0 .o 0.0

U .3

U.O

0. . J 0 a > i O.bd 14 7 0.310 7J 0. 1 5 7 4 1 0 .3L*I 3 0.222 5 0 0.050 I L 0.L74db G.U30J3 0.31468 0.03 996 0.34234 0.04257 0.35857

0.1,

3. 3 1 3 L L 0.13Jttl

J.31841 J. 1 4 0 4 2 J .J 5 0 1 5 0. 2cIu77 3.0 2 2 5 1 0.35515 d.Ubbjj 0.42454 *i.3 I 7 7 1 J.41321 3.0db0d J.3 J 5 1 ~ 0.UY 1 2 0 0.52315

0.O431d 0.36392 0.0423 I 0.35851 0.03996 0.34234 U.d3693 0.31 4 6 8 0.03072 0.2748 7 0.U241Y 0.22250 U.Ul670 0.15741 0.00852 0.08187 0 .u 0.0

I).J&>d.i 0.2JJYI 0.U3

I.tY

0.2 7 Y d l

0.04I5d U-5413U d.33) 1 9 0 . 5a036 U.0tld‘t 0.41 bY3 J.Ub>5> 0.43435 0. Ubo79 0.44333 0.06355 0.43-55 0.06184 0.41oY3 0.055 19 0.3dt-38 3 . &750 J.3415U 3.u3749 U.27Yd7 0.02569 O.LU0Y7

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J. O Y L Y 2

3.52895

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0.35512

0.036 1 5 0. ZbOY 7 0.01d47 9.14042 0.u

u.u

u.

J.3

0.0

0.0

3.1,

d.U

d

J.lJ

0.U

3 . 2 JLUb

11.1)+31* 3.4YLt3Q

1

ij. u o l a J 1,.‘i73LL

U. J / 6 + 0 J .Orto34

1-0uudJ

0.11L3‘t J. 7 7 5 0 1

3.1+101 1. U U U U I l

J.U>l”+

J. U U J I d

3.>Y1Iu ii. 112 I 2 J. 3 2 9 8 b

J.l>Ao>

d. 7 0 1 3 ~ 3.1-r~So 3.7Lb43 0. l 5 j d b 0. 7 5 9 Ii 0.1500 7 0 . 743Y I 0.1>300 0. 7 5 9 1 7 0.14540 0.12044 0.131b2 J. 7 0 1 5 0 0.11272 0. b 5 Y d b 0.UdYId 3. 5 9 1 7b

0.0018d u.+rwL d.03104 O.LY2Ub 0. U 0.3

U.

J. i r ~ d o a 3.

0.19tJ54 I

0.dOBA

b.IY31d 9.d65d4 U. 1 d L d L 0.82833 0.1b2dl 0.d4394 3.14235

0.81tilU U.11294 0.77301 0.0 Illuo 0.6a634 0. U4UL4 0.49456 0.0 U.U

ufd9U

U.173311

14L3~

3 . d 16 / J 3. I o 5 d I 0.04594 u. l d L d 2 J.63d33 0. l Y 3 1 J i).d0584

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1 .3U0UU

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1 .0U0dU

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1. dJUUU

0.24345

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1.

0ouuii

0.176UJ 1. 000ou U.141dL 1.0000u 0. 0 Y 8 9 0 1 .00000 U-O>Udb 1.00000 0.0 0.0

Author Index

A Akheiser, N. I., 35 Ambarzumian, V. A., 85 Angel, E., 60, 86, 87, 88, 117, 118, 136, 137, 161, 180 Azen, S. P., 143

B

Beckenbach, E. F., 179 Bellrnan, R., 11, 21, 26, 35, 36, 59, 60, 86, 87, 88, 117, 118, 136, 143, 161, 179 Bergrnan, S., 88 Birkhoff, G., 137 Bramble, J. H., 35 Bremrner, H., 161 Buzbee, B. L., 117, 118, 136

C Carnahan, B., 136 Casti, J., 36, 143, 180 Chandrasekhar, S., 85

Chernin, K. Y.,60 Cherry, I., 143 Callatz, L., 36 Collins, D. C., 136, 137 Cooke, K. L., 143 Cornock, A. F., 86 Courant, R., 35 Crank, J., 161 Cuthill, E. H., 137

D Denman, E. D., 117 Detchmendy, D., 180 Distefano, N., 60, 87, 88, 117 Dorr, F. W., 60, 86, 118, 136 Douglas, J., Jr., 137

F

Fomin, S. V., 35, 86 Forsythe, G. E., 136 201

202 G Gelfand, I. M., 35, 86 George, J. A,, 118 Godunov, S. K., 86 Golub, G. H., 117, 118, 136 H Hickerson, N., 87 Hilbert, D., 35 Hockney, R. W., 136 Hubbard, B. E., 35 Huss, R., 88 I Isaacson, E., 36, 161 Jain, A , , 87 Jong, Dah-Teng, 142 Juncosa, M., 179

K Kagiwada, H., 86, 180 Kalaba,R., I I , 86,87,88,161,179,180 Kantorovich, L. V., 36, 60 Karlquist, O., 86 Kashef, B., 143 Kato, H., 21 Keller, H. B., 36, 161 Knuth, D. E., 21 Krylov, V. I., 36, 60 L Lax, P., 161 Lanczos, C., 161 Lehman, R. S., 21, 87 Lew, A,, 137 Lockett, J., 11, 161 Luther, H. A,, 136 Lynch, R. E., 136

M McNabb, A., 88, 118 Maynard, C., 87 Meyer, G. H., 86 Mikhlin, S. G., 117 Morton, K. W., 36, 161 N Nicolson, P., 161 Nielson, C. W., 117, 136

Osborn, H., 60

0

P

Peaceman, D. W., 137, 161

R

Rachford, H. H., Jr., 137, 161 Ralston, A., 161 Redheffer, R., 117 Reid, W. T., 117 Rice, J. R., 136 Richardson, J. M., 143 Richardson, L. F., 60 Richtmeyer, R. D., 36, 161 Rosser, J. B., 87 Ryabenki, V. S., 86 Rybicki, G. B., 86 5 Sage, A. P., 60 Schiffer, M., 88 Schujman, J., 117 Schurnitzky, A., 88, 118 Scott, M., 87 Srnolitskly, K. L., 117 Soillers, W. R., 87

T Todd, J., 36, 59, 60, 136. 161 Thomas, D. H., 136 Thornee, Vidar, 35 Usher, P. D., 86

U

V Van der Pol, B., 161 Varge, R., 35, 36, 60, 117, 136, 137 Von Rosenberg, D. U., 87

w

Wasow, W. R., 136 Wilf, H. S., 36, 161 Wilkes, J. O., 136 Williamson, J., 136 Wing, G. M., 86, 143 Young, D., 137

Y

Subject Index

A Alternating-direction implicit methods, 134, 153 B Biharmonic equation, 60, 80, 103 Block iterative methods, 132 Bubnov-Galerkin method, 10 Building block technique, 60

C

Courant parameter, 1 1

D Deferred passage to the limit, 49 Diagonal decomposition, 126 Differential inequality, 166 Differential quadrature, 36, 142 Discretization, 28 Disturbed control, 57 Dynamic programming, 12,40

E

Efficiency, 47 Existence and uniqueness, 8 Explicit methods, 148

F

Finite difference methods 136 Functional equation, 13

G Gaussian quadrature, 157 General regions, 114 Green’s function, 24 H Heat equation, 144 Higher order equations, 55

I

Implicit methods, I51 Invariant imbedding, 72, 86, 87, 138 Inversion of Laplace transform, 158 Irregular regions, 52, 89 Iterative analysis, 60 203

204 K

Kroncckcr product, 121 Kronecker sums, 122

L Laplace transform, 156 M Method of lines, 36 Minimum convolution, 17 N Newton-Raphson-Kantorovich method, 173 Nonlinear equations, 162 Numerical analysis, 59

P Parabolic equations, 144 Point iterative methods, I28 Poisson equation, 86 Potential cquation, 22 Principlc of optimality, 14 Properly posed problems, 146

R

Random walk, 82 Rayleigh-Ritz method, 10 Riccati equation, 71 Riccati transformation, 62

S Semidiscretization, 36 Sensitivity equations, 174 Single sweep methods, 64 Sparse symmetric systems, 87 Stability, 44 Successive overrelaxation method, 130

T Tensor product, 136 Three-dimensional equations, 101 Tridiagonal matrices, 20

U

Unconventional difference methods, 138

Q

Quadratic convergcnce, 169 Quadratic variational problcms, 6 Quasilinearization, 162, I79

V Variational approach, 7

Mathematics in Science and Engineering A Series of Monographs and Textbooks Edited

by RICHARD BELLMAN, University of Southern California

23. A. Halanay. Differential Equations: Stability, Oscillations, Time Lags. 1966

42. W. Ames. Nonlinear Ordinary Differential Equations in Transport Processes. 1968

24. M. N. Oguztoreli. Time-Lag Control Systems. 1966

43. W. Miller, Jr. Lie Theory and Special Functions. 1968

25. D. Sworder. Optimal Adaptive Control Systems. 1966

44. P. B. Bailey, L. F. Shampine, and P. E.

26. M. Ash. Optimal Shutdown Control of Nuclear Reactors. 1966

27. D. N. Chorafas. Control System Functions and Programming Approaches (In Two Volumes). 1966 28. N. P. Erugin. Linear Systems of Ordinary Differential Equations. 1966 29. S. Marcus. Algebraic Linguistics; Analytical Models. 1967 30. A. M. Liapunov. Stability of Motion.

Waltman. Nonlinear Two Point Boundary Value Problems. 1968

45. lu. P. Petrov. Variational Methods in Optimum Control Theory. 1968 46. 0. A. Ladyzhenskaya and N. N. Ural'tseva. Linear and Quasilinear Elliptic Equations. 1968

47. A. Kaufmann and R. Faure. Introduction to Operations Research. 1968

48. C. A. Swanson. Comparison and Oscillation Theory of Linear Differential Equations.

1966

1968

31. G. Leitmann (ed.). Topics in Optimization. 1967

49. R. Hermann. Differential Geometry and the Calculus of Variations. 1968

32. M. Aoki. Optimization of Stochastic Systems. 1967

50. N. K. Jaiswal. Priority Queues. 1968

33. H. J. Kushner. Stochastic Stability and control. 1967 34. M. Urabe. Nonlinear Autonomous Oscillations. 1967 35. F. Calogero. Variable Phase Approach t o Potential Scattering. 1967 36. A. Kaufmann. Graphs, Dynamic Programming, and Finite Games. 1967

37. A. Kaufmann and R. Cruon. Dynamic Programming: Sequential Scientific Management. 1967 38. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh. The Theory of Splines and Their Applications. 1967

39. Y. Sawaragi, Y. Sunahara, and T. Naka-

51. H. Nikaido. Convex Structures and Economic Theory. 1968 52. K. S. Fu. Sequential Methods in Pattern Recognition and Machine Learning. 1968

53. Y. L. Luke. The Special Functions and Their Approximations (In Two Volumes). 1969 54. R. P. Gilbert. Function Theoretic Methods in Partial Differential Equations. 1969 55. V. Lakshmikantham and S. Leela. Differential and Integral Inequalities (In Two Volumes). 1969

56. S. H. Hermes and J. P. LaSalle. Functional Analysis and Time Optimal Control.

1969 57. M. Iri. Network Flow, Transportation, and Scheduling: Theory and Algorithms. 1969

mizo. Statistical Decision Theory in Adaptive Control Systems. 1967

58. A. Blaquiere, F. Gerard, and G. Leitmann. Quantitative and Qualitative Games.

40. R. Bellman. Introduction to the Mathematical Theory of Control Processes, Volume I. 1967;Volume II. 1971 (Volume Ill in preparation)

59. P. L. Falb and J. L. de Jong. Successive Approximation Methods in Control and Oscillation Theory. 1969

41. E. S. Lee. Quasilinearization and Invariant Imbedding. 1968

60. G. Rosen. Formulations of Classical and Quantum Dynamical Theory. 1969

1969

62. R. Bellman, K. L. Cooke. and J. A.

79. M. A. Aiserrnan, L. A. Gusev, L. 1. Rozonoer. 1. M. Smirnova, and A. A. Tal'. Logic, Automata, and Algorithms. 1971

1970

80. Andrew P. Sage and James System Identification. 1971

63. E. J. Beltrami. An Algorithmic Approach t o Nonlinear Analysis and Optimization. 1970

81. R. Boudarel, J. Delmas, and P. Guichet.

61. R. Bellman. Methods of Nonlinear Analysis, Volume I. 1970 Lockett. Algorithms, Graphs, and Computers.

64. A. H. Jazwinski. Stochastic Processes and Filtering Theory. 1970 65. P. Dyer and S. R. McReynolds. The Computation and Theory of Optimal Control. 1970 66. J. M. Mendel and K. S. Fu (eds.). Adap-

L. Melsa.

Dynamic Programming and I t s Application t o Optimal Control. 1971

82.F. V. Atkinson. Multiparameter Eigenvalue Problems, Volume I. 1972 83.William S. Meisel. Computer-Oriented A p proaches to Pattern Recognition. 1972

tive, Learning, and Pattern Recognition Systems: Theory and Applications. 1970

84. M. Frank Norman. Markov Processes and Learning Models. 1972

67. C. Derman. Finite State Markovian Decision Processes. 1970

85. G. E. Ladas and V. Lakshmikantham. Differential Equations in Abstract Spaces. 1972

68. M. Mesarovic, D. Macko, and Y. Taka-

86. William T. Reid. Riccati Differential Equations. 1972

hara. Theory of Hierarchial Multilevel Systems. 1970

69. H. H. Happ. Diakoptics and Networks. 1971 70. Karl Astrom. Introduction t o Stochastic Control Theory. 1970 71. G. A. Baker, Jr. and J. L. Gammel (eds.). The Pade Approximant in Theoretical Physics.

87. Bruce A. Finlayson. The Method of Weighted Residuals and Variational Principles: With Application t o Fluid Mechanics, Heat and Mass Transfer. 1972 88. Edward Angel and Richard Bellman. Dynamic Programming and Partial Differential Equations. 1972

1970 72. C. Berge. Principles of Combinatorics. 1971

In preparation

73. Ya. 2. Tsypkin. Adaptation and Learning in Automatic Systems. 1971

Alexander Weinstein and William Stenger. Methods of Intermediate Problems for Eiganva1ues:Theory and Ramifications

74. Leon Lapidus and John H. Seinfeld. Numerical Solution of Equations. 1971

Ordinary Differential

75. L. Mirsky. Transversal Theory. 1971 76. Harold Greenberg. Integer Programrning. 1971 77. E. Polak. Computational Methods in O p timization: A Unified Approach. 1971 78. Thomas G. Windeknecht. General Dynamical Processes: A Mathematical Introduction. 1971

G. Arthur Mihram. Simulation: Statistical Foundations and Methodology Renfrey B. Potts and Robert M. Oliver. Flows in Transportation Networks Umberto Bertele and Franchesco Brioschi. Nonserial Dynamic Programming H. Melvin Lieberstein. Theory of Partial Differential Equations H. M. Nussenzveig. Causality and Dispersion Relations