EXISTENCE OF ALGEBRAIC MATRIX RICCATI-EQUATIONS ARISING IN TRANSPORT-THEORY

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N O R T H - ~

Existence of Algebraic Matrix Riccati Equations

Arising in Transport Theory

Jonq

Juang*

Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan

Submitted by Richard A. Brualdi

ABSTRACT

We consider the existence of positive solutions of a certain class of algebraic matrix Riccati equations with two parameters, c (0 ~< c < 1) and a (0 < a ~< 1). Here c denotes the fraction of scattering per collision, and a is an angular shift. Equations of this class are induced via invariant imbedding and the shifted Gauss- Legendre quadrature formula from a simple transport model. By establishing the existence of positive solutions of such equations, the problem of the convergence of some iterative schemes for solving them can be completely solved.

1. I N T R O D U C T I O N

C o n s i d e r t h e algebraic matrix Riccati e q u a t i o n o f t h e form

B - A S - SD + SCS = 0 . ( 1 )

* The work was partially supported by the National Science Council of R.O.C.

LINEAR ALGEBRA AND ITS APPLICATIONS 230:89-100 (1995)

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H e r e A, B, C, and D are matrices having the following structure:

1

1 ]

A N - × N - = diag c ( w [ + or) ' c ( w ~ + a ) ' ' C ( W N - + a )

-

Z(w;+ ~ ) ' 2 ( w ; + ~ ) ' ' 2(w~-~ ~)

:= D A - ia T,

[1]

where i = " ;

i

1

1 ]

ON+ xN += diag c ( w f - a ) ' c ( w ~ - a ) ' ' c ( w ~ + - a ) C+ 1 C2 + CN+ + ] i 7 - , , ,

2 ( w 1 - o,) 2 ( w ; - o,) 2 ( w ? , + - o,) := D D - idT;

B = iiT; C ~ da T.

Equation (1) contains two parameters c and or. H e r e c denotes the average total n u m b e r of particles emerging from a collision, which is assumed to be conservative, (i.e., 0 ~< c ~< 1), and a (0 ~< a ~< 1) is an angular shift. T h e dimensionally d e p e n d e n t quantities w i- and w~ + denote the Gauss- L e g e n d r e sets (see, e.g., [12]) on [ - a , 1] and [ or, 1], respectively; and ci- and c + are, respectively, their corresponding weights. Without loss of generality, we shall assume that

- a < w f < w ~ < ... < w ; - < l and a < w ~ < w $ < ... < w • + < l . Such an equation is induced via invariant imbedding (see, e.g., [1, 2, 15]) and the shifted Gauss-Legendre quadrature formula from a simple transport model [5, 6].

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ALGEBRAIC MATRIX RICCATI EQUATIONS 91 For ot = 0, two iterative procedures for finding the minimal positive solution (in the componentwise sense) of Equation (1), one corresponding to a nonlinear version of the Gauss-Jocobi (GJ) method and the other associated with a nonlinear version of the Gauss-Seidel (GS) method, were proposed, respectively, by Shimizu and Aoki [13] and by Juang and Lin [9]. While such iterative procedures have been proved quite effective in practice (see [10-11] and the work cited therein), their convergence has not yet been fully investigated. Sufficient conditions for convergence of the GJ and GS methods were given in [10] and [9], respectively. However, it was noted (see Table 2 of [8]) that those sufficient conditions will fail if c is not far away from 1. And it was also observed (see Theorem 1 of [8]) that the existence of a positive solution of (1) implies the convergence of both iterations. This observation can be easily extended to the case that a ~ 0. Therefore, to completely solve the convergence problem one needs to find a direct method for establishing the existence of positive solutions of Equation (1) for all 0 ~< c ~< 1 and 0 ~< ot ~< 1. This is what motivates our work here.

In this article, we first show that an

a priori

bound, which is independent of c and a, can be obtained by introducing a one-parameter (kl, 0 < k 1 < 1) family. Therefore, degree theory is applied to show the existence of positive solutions. Some applications and concluding remarks are given in Section 3.

2. MAIN RESULTS

To derive our main results, we first write Equation (1) in the component form

- - + - S q = c 1 + 1 + .

W/-q- O/ W ; - - O~ 2 k=l Wk "q- O~ 2 = W~-- O~

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The structure of (2) suggests that we seek a solution of the form

+

(4)

where

JONQ JUANG

Define

and

N - -

c •

c k

(6a)

x = 2k=1 hk

c ~ c;

(6b)

Y = 2 k=l-~Z

1 N + +

k--~l

Ck

Sa'

1 ~< i ~< N-,

(3b)

h i = l +

2 = w ~ - - a

1 ~

c[

lj = l + -~ k=l w---~ otSkj, I <~j <~ N +.

(3c)

Substituting (3a) into'(3b) and (3c), respectively, we obtain

c N+ c~(w~-+

~)

h, = 1 + -~ ~_, w~-+ w~ h'lk' 1 <~ i <~ N - ,

(4a)

k = l

c ~ c[(w 7 - or)

N +.

(4b)

lj = 1 + 7

k : l

W--"~TW5

hklj' 1 <~ j <~

Set h~ = l/h~,

lj

= 1/~; then Equation (4) can be equivalently reduced to

N + + c c

h , = 1 - 7

= "~'-k +

,

l < ~ i < ~ N - ,

(5a)

9 k:l ( w , + w;)fk

c ~ c;

c[__E

' ( w ; + ~ ) c ;

N + (Sb)

~ = 1

2 k=X "~-'k-k q'-7 =

(w;+w/.)[tk,

l<<.j4

.

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ALGEBRAIC MATRIX RICCATI EQUATIONS 93 Multiplying (5a) b y cc7/2[~i, and s u m m i n g the resulting equation over the index i, we have that

2 i = 1

2 c[

=: x - xy + a. ( 7 a )

A similar p r o c e d u r e is applied to (5b) to get

2EC+--i=I

2 =y--xY+-21k=X'~k ~2i=l(Wi-+w~)h i

= : y - x y + b (7b)

Adding (7a) and (7b), we obtain

( 1 - x ) ( 1 - y ) = l - c .

₍₈₎

REMARK.

1. F o r a = 0, the quantities h i and lj are the descrete version of Chandrasekhar's well-known H functions [3, 4].

2. F o r a = 0, (8) reduces to a descrete version o f s o m e expressions [3, 4] concerning the properties o f H functions.

Since a + b = xy, we see immediately, for a ~: 1, that if h i and lj are positive solutions o f (4), t h e n there m u s t exist two positive n u m b e r s k I and k 2 , w h e r e 0 < k 1,k 2 < l a n d k 1 + k 2 = 1, s u c h t h a t

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It then follows from (7), (8), and (9) that the following holds: 1 - ( c / 2 ) ( 1 - a ) + k l c ± ¢ [ 1 - ( c / 2 ) ( 1 - or) + k l c ] 2 - 2k,(1 + a ) c x = 2k 1 = : a 1 ± b 1, (10a) 1 - ( c / z ) O + ~ ) + k2c ± v/[1 - ( c / 2 ) 0 + ~ ) + k2~] 2 - 2k2(1 - ~ ) ~ y = 2k 2 = : a z ± b 2. (10b)

Since k 1 and k 2 are to be treated as real parameters, necessary conditions for (10) to be meaningful are that both [1 - ( c / 2 ) ( 1 - a ) + k l c ] 2 - 2k~c(1 + a ) and [1 - ( c / 2 ) ( 1 + a ) + k 2 c ] 2 - 2 k 2 c ( 1 - a ) are nonnegative. How- ever, these are so if 0 ~< a ~< 1 and 0 ~< c ~< 1. To see this, we note that, for

c ~ O, f l ( k l ) := [1 - ( c / 2 ) ( 1 - a ) + k l c ] 2 - 2 k l c ( 1 + a ) has a m i n i m u m (1 + a ) ( 1 - or)(1 - c), which is nonnegative w h e n e v e r 0 ~< a ~< 1 and 0 ~< c ~ < l .

W e denote by F the feasible region {(k, c, or) : 0 < k < 1, 0 ~< c ~< 1, and 0 ~< ~ < 1} for the solution o f (1). T h e properties and signs o f 1 - x and 1 - y will be examined in the next lemma.

LEMMA 1.

(i) 1 - a 1 + b 1 >~ 0 a n d 1 - a 1 - b 1 <~ 0 f o r all ( k l , c, or) ~ F.

(ii) 1 - a 2 + b 2 >~ 0 a n d l - a 2 - b 2 <~ O f o r all ( k 2 , c, a ) ~ F . 1 1 T h e n 1 - a I + b 1 >1-~ (iii) L e t c b e s u f f i c i e n t l y s m a l l , s a y 0 <~ c <~ ~. 6 a n d l - a 2 + b 2 >t ~ f o r a l l k l a n d k 2 , 0 < k 1, k 2 < 1, a n d a l l a , O <~ a <~ 1.

P r o o f . Since the computation leading to (i) and (ii) is similar, we shall only prove (i). To see (i), it suffices to show that bl z > / ( 1 - al )2, or equivalently

[ c

]2

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ALGEBRAIC MATRIX RICCATI EQUATIONS 95 Since the left-hand side o f the inequality is equal to 4(1 - k l ) ( k l ) ( 1 - c), the assertion o f L e m m a 2(0 thus follows.

1 T o p r o v e (iii), w e see that if 0 ~< c ~< g, then a i -- b 1

(1 +

1 - @ / 2 ) ( 1 - a ) + k~c + ¢ [ i - @ / 2 ) ( 1 - ~ ) + k l c ] 2 - 2 k l c ( l + a ) 1

< 2c(1 + a ) <

for all k 1 and a . Thus, 1 - a 1 + b I >/ ½, as asserted. Similarly, we have a 2 -- b 2 (1 - a ) c 1 - @ / 2 ) ( 1 + a ) + k2c + ¢ [ 1 - @ / 2 ) ( 1 + a ) + kzc] z - 2k2c(i - a ) c c 1 ~< 1 - @ / / 2 ) ( 1 + a ) ~< ~ ~< -7" T h e r e f o r e , 1 - a 2 + b 2 >~ -~, as asserted. • In view o f (8), we see that if h~ and lj are solutions o f (4), then either

1 - x > ~ 0 and 1 - y > ~ 0 ( l l a ) o r

1 - x < 0 and 1 - y < 0 . ( l l b ) An a priori bound, which is i n d e p e n d e n t o f k l , c, and or, is obtained in the following l e m m a .

LEMMA 2. Let fh a n d [j be a n y positive solutions o f (5) satisfying ( l l a ) . T h e n t h e re is an m > 0 such that min{h i, lj} >~ m f o r all i, j , all 0 <<. a <~ 1, a n d all 0 <~ c <~ 1.

Proof. Using (4), w e see clearly that /~i ~< 1 and /j ~< 1 for all i , j . T h e r e f o r e ,

N + c ( w : -

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a n d

C ~

W k C k

/J ~> 1 - x + 2 k = 1 W k q - 1"

S i n c e 1 - y > / 0 a n d 1 - x > / 0 , t h e r e m u s t exist positive c o n s t a n t s k 1 a n d k e, k I + k e = 1, s u c h t h a t 1 - x = 1 - a 1 + b I a n d 1 - y = 1 - a e + b e, w h e r e a I - b I a n d a e - b e are d e f i n e d in (10). Now, via L e m m a l(iii),

1 f o r 0 ~<c ~< 1

1 - x >/ ~ ~. C o n s e q u e n t l y ,

1 ~ w k c k

>i m i n 1 , ]-6 ( w [ + 1) =: m2 > 0

k = l

for all j , all 0 ~< c ~< 1, a n d all 0 < ot ~< 1. O n t h e o t h e r h a n d ,

N+ + ~ ( 1 -

,~)

C C k Y = 2 k ~ l -~k ~ 2 m 2 a n d so 1 - y > / 1 - c(1 - a ) / 2 m 2. H e n c e , if 0 ~< c ~< m 2 o r a >/ 1 - m 2, 1 t h e n l - y >/ 7. H o w e v e r , i f l > ~ c > i m 2 a n d 0 ~ < a ~ < 1 - m 2 , t h e n N + N + c ( w ~ - a ) c [ m 2 ( w ~ - 1 + m2)c ~ = k = l 1 + w k C o n s e q u e n t l y , f~i /> min{½, m l } : = m l - T h e a s s e r t i o n o f t h e l e m m a n o w follows o n c h o o s i n g m = m i n { m 1, m2}.

THEOREM 1. Equation (1) has positive solutions satisfying ( l l a ) for all O <~ c <~ 1 and O < a <~ 1.

Proof. U s i n g (3a), L e m m a 2, a n d t h e fact t h a t wj + >/ot a n d w f >1 - a

for all i, j , w e c o n c l u d e t h a t for a n y positive s o l u t i o n Sij o f (2) satisfying ( l l a ) ,

c ( 1 - a 2) m 2 m e ~ - -

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ALGEBRAIC MATRIX RICCATI EQUATIONS 97 for all i , j , all 0 ~< c ~< 1, and all 0 ~< ot ~< 1. H e r e m is defined as in L e m m a 2. Let S be a column vector defined as

= ( a l l , . . . , S I X + , $21, $22 ,°°" , S 2 N + , ' " , S N - I , " " ,

aN-N+) T.

Then (2) can be formulated as

g = Fc(g),

where Fc is a continuous and nonlinear map from a N-N+ t o R N-N÷ . Choose

(

)(

= - - - - 1 + ~ > 0 ,

r 1 + 4 k = l W k + O l 4 = w [ - o t

and let D = {x ~ R n-n+ : Ilxll~ < r}. Clearly, D is a bounded open set in R N-u+, and F c is continuous on D. Consider the homotopy H~ = I - F c, and suppose that S - Fc(S) = 0 for g ~ D; then IISIl~ =

IIFc(S)ll~ <

r / 2 < r. Thus S ~ D. Hence, by homotopy invariance (see, e.g., T h e o r e m 13.2.11(ii)

of [7]),

d ( H c , 0 , D ) = d ( H o , O , D ) = d ( I , 0 , D ) = 1.

The above argument is true for all 0 ~< ot ~< 1. Therefore, we conclude that Equation (1) has positive solutions for all 0 ~< c ~< 1 and 0 ~< a ~< 1. •

REMARK. We are motivated by the work of Stuart [14] to use the homotopy argument to show the existence of positive solutions of (1).

3. A P P L I C A T I O N S A N D C O N C L U D I N G REMARKS

As in the case that a = 0 (i.e., no angular shift), the iterative procedures for solving the minimal positive solution of the equation (1) can be classified into three types: first, the iteration of Aoki and Shimizu, which is essentially a nonlinear version of Ganss-Jocobi (GJ); second, the iteration of Juang and Lin, which is essentially a nonlinear version of Ganss-Seidel (GS); third, a nonlinear version of SOR, whose effectiveness has yet to be studied theoreti-

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eally. We now define the GJ and GS methods, respectively, as follows:

(

t~ k O k j

)(

_ _ _ _

)

S}? +1) = c ( w i - - t - o l ) ( w ? o l ) 1 ~ - ~ ( P ) 1 c ; S } f f )

w ; + w ?

1+ 7~=, w-~7-~

1 + 7 = w ; - ~

(12a)

S,tj °) = 0 for all i, j, (12b) and

(

)

= _ L 2 J _

w ; + w [

1 +

2k:~

w ; + ~

+ 2k:~w[+~

× 1 + ~ w~--~_~ + ~ = . w - - - ~ ; , ( l a a ) ~ ) = 0 for all i, j. (13b) Let S = ( S ? ) be a positive solution of Equation (1), whose existence is assured by Theorem 1.

An easy induction will give

max(S~f), S~j p)} ~ S q for all i, j and all p.

It is also clear that for each i. ". the iterations IS(P)/~ - and f.q(P)I~ _{- J -} _{" i j "p = U} _{" - - i ]} _{"p = 0 ~ * ~}~,~ monotonically increasing. Therefore, the limits of both iterations exist; .they

(~) (~) (~) (~)

will be denoted by Sq and S i, , _(~)respectively. Furthermore, ( S i , ) = _(~) _s ( S i j ) .

To see this, we first note that ~S? ) and ( S I j ) both are positive solutions of (p) ( )

(1). Therefore, an reduction will gave Sq <<. Sij for all I, j, and p. Thus, S~;' ~< S~;). Similarly, S~; )/> S~;'. We summarize the above results as follows. THEOREM 2. _M _~ For all 0 ~ c <~ 1 and 0 <<. ~ <~ 1, the iterations _. _. _. _. _. {S(P)I ~ _{U "p = o}

and {S~P)}p=0 converge to the minimal postttve solution Smi n o f (1).

The minimal positive solution Smi n o f (1)/s defined in the following sense: if S is a positive solution of (1), then S >~ Smi,, i.e., S,j >1 (Smi,),j f o r all i , j .

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ALGEBRAIC MATRIX RICCATI EQUATIONS 99 THEOREM 3. Equation (1) has a unique positive solution satisfying ( l l a ) for O <~ c <<, 1 and O <~ ~ <<, 1.

Proof. Let (hmin) i and (lmi,) j be the minimal positive solution pair of (4), whose existence assured by T h e o r e m 2. Let X mi n and Yrnin be defined as in (6) except that h i and lj are replaced by (hmin) i and (lmin)j, respectively. Consequently, Xmi . ~< x and Ymin ~< Y" On the other hand, for c ~ 1, 1 - Xmi n = ( 1 - - { 7 ) / ( 1 - - Y m i n ) ~ ( 1 - - ¢ ) / / ( 1 - - y ) = 1 - - X. Hence, Xmi n = X, SO that h i = (hmin) i for all

i. Similarly,

l, = (lmin) j. The uniqueness of the positive solution of (1) satisfying ( l l a ) now follows from (3a). •

We conclude this paper by suggesting the following further related matters:

1. It would be interesting to study the bifurcation diagram of positive solutions of Equation (1) as c and a vary from 0 to 1.

2. It is of interest to investigate the effectiveness of the NSOR.

3. Additional complexities of the reflections or scattering problem can also be considered, such as anisotropic scattering, spatially distributed sources, and time dependence.

REFERENCES

1 R. E. Bellman, R. E. Kalaba, and C. Prestrud, lnvariant Imbedding and Radiative Transfer in Slabs of Finite Thickness, Amer. Elsevier, New York, 1963. 2 R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding, Wiley,

New York, 1975.

3 I.W. Busbridge, The Mathematics of Radiative Transfer, Cambridge U.P., New York, 1960.

4 S. Chandrasekhar, Radiative Transfer, Dover, New York 1960.

5 F. Coron, Computation of the asymptotic states for linear half space kinetic problem, Transport Theory Statist. Phys. 19(2):89-114 (1990).

6 B.D. Ganapol, An investigation of a simple transport model, Transport Theory Statist. Phys. 21(1 & 2):1-37 (1992).

7 V. Hutson and J. s. Pyre, Applications of Functional Analysis and Operator Theory, Academic, New York, 1980.

8 J. Juang and I-Der Chen, Iterative solution for a certain class of algebraic matrix Riccati equations arising in transport theory, Transport Theory Statist. Phys. 22(1):65-80 (1993).

9 J. Juang and Z. T. Lin, Convergence of an iterative technique for algebraic matrix Riccati equations and applications to transport theory, Transport Theory Statist. Phys. 21(1 & 2):87-100 (1992).

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10 P. Nelson, Convergence of a certain monotone iteration in the reflection matrix for a nonmultiplying half-space, Transport Theory Statist. Phys. 13:97-106 (1984).

11 P. Nelson, A. K. Ray, and C. Garth, in Advances in Reactor Computations, Vol. II, Amer. Nucl. Soc., 1983, p. 623.

12 A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, McGraw- Hill, New York, 1978.

13 A. Shimizu and K. Aoki, Application of Invariant Embedding to Reactor Physics,

Academic, New York, 1972.

14 C. A. Stuart, Existence theorems for a class of nonlinear integral equations,

Math. Z. 137:49-66 (1974).

15 G.M. Wing, An Introduction to Transport Theory, Wiley, New York 1962.