Preconditioned method in parallel computation1

Journal of Systems Engineering and Electronics, Vol . 1 7 , No. 1 , 2006, p p . 220 -222

Preconditioned method in parallel computation* Wu Ruichan & Wei Jianing School of Science, Wuhan Univ. of Technology, Wuhan 430063, P. R. China (Received December 28, 2004)

Abstract: The grid equations in decomposed domain by parallel computation are soled, and a method of lccal orthogonaization to solve the largescaled numerical computation is presented. It constructs preconditioned iteration matrix by the combination of predigesting LU decomposition and lccal orthogonalization, and the convergence of solution is proved. Indicated from the example, this algorithm can increase the rate of computation efficiently and it is quite stable. Key words: grid equations, parallel computation, precondition, LU decomposition, local orthogonalization.

1. INTRODUCTION 2. THEOREM AND ALGORITHM Decreasing the times of communication and increasing the rate of it in the whole domain is very important For given equations for improving the rate of parallel computation in deAx=b (1) composed domain. Preconditioned technology has where A is invertible square matrix, and A E RN N , been used widely in this arear1]because it can change x , b E R N . Let B is invertible square matrix of orthe mode of communication and improve the topologider N , then the preconditioned iteration of Eq. (I) is cal configuration. Orthogonalization is a perfect preconditioned technology, but it is impossible to orthogonalize entirely for the large workload of computation use block LU decomposition to Eq. (1) as LUX = b , and effect of rounding error. So, local orthogonalizaand we set tion is presented and the computation can be simpli'u11 u 1 2 0 ' fied by this method. Predigesting LU decomposition U" = .*Um-lrn that belongs to an incomplete LU decompositi~n[~-~l ,o Umm , is actually a method of Gauss eliminatio~$~~~]. In the

I

Ref. [ 3 , incomplete LU decomposition with partial fill-in ( ILU ( e ) ) is suggested, but the quantitative analys of the parameter is difficult. LU decomposition with spectral factorization is presented in Ref. [ 4 1 , this method is not fit for the decomposition of single matrix but for abundant matrixes with the same nonzero-configuration. Furthermore, the errors given

by this method must be considered. Some properties and proofs about the elimination- preconditioned method given in Refs. [ 5 , 6 ] are theoretic bases of the predigesting LU decomposition. Preconditioned itera-

where LR is residual matrix in the Eq. ( 3 ) , it can be adjusted for LU * Z A . Theorem 1 The preconditioned iteration (4) is converged with LU * Z A .

Prove The iteration matrix of Eq. (4) is

tion matrix is constructed by the combination of predigesting LU decomposition and local orthogonalization, by use this method, the rate of convergence is obviously improved.

*

where U * - R is uptriangle form and the submatrix of main diagonal is zero matrix. SO p( U * -'R) = O .

This project was supported by the National Natural Science Foundation of China (70371063, 60173046)

221

Preconditioned method in parallel computation is A1 and A2

Use transform x = Qy to Eq. (1), where Q is transform matrix, it can orthogonalize the corresponding sub-matrix of U " . Let U * Q = C , then ,- ,

'..

C= I

Cm-lm vmm

W

(5)

.

'..

514

20x20

---

and b2 = ( 1 ,1 , 1 )zoox

bl = ( 1 ,1 ,em-1)zO

1

respec-

Table 1 Comparative result of iteration of Gauss-seidel and algorithm 2.2

,

Iteration of Gauss-Seidel

Imm

(7)

,

I .

RmxXm3,i= 1 , 2 , . - - , m , ~ m = i N . i=l

Theorem 2 If y(') converges to y * in Eq. (61, then, x ( ~also ) converges to x * in (4). IIx(')-x* II = II ~ y (-~~ ) y II *= Prove

11 Q(dk' - y*1 11 < 11 Q 11 11 Y")

(1

-Y *

11

11 Q 11

is limitary, x(') - x * II +o.

Algorithm

( 1 ) Giving initial iteration value y(O) € R" , ern ror limitation E , minimum iteration count N ~, integer k = O ; ( 2 ) If k > N ~ n go , to step (4); otherwise calculate y('+') from(6) ; (3) If I y('+l) - y(') I e , go to step (4); otherwise y ( k + ' ) + y ( ' ) go to setp ( 2 ); (4) Based on x('+')= Q Y ( ~ + ' )we , can obtain the iteration solution of Eq. (1); (5) Output numerical result; stop.

<

In addition, if we set * = diag ( Uii , Uzz, * * * U m m ) ,this decomposition is still a predigestion of

LU decomposition, but the iteration result is worse than U * (Table 1 ) .

3. NUMERICAL RESULIS AND CONCLUSION For given linear equations Ax = b ,mfficient matrix

I

4x4 4x4

I

I

Iteration of

I

Convergence

I

4

I

40x40 40x40

I

50 27

I

I Convergence

I I

I

Gauss-Seidel

LU"

m I..

where Q is transform matrix, and for I( y(') - y * 11 +O, so

Divergence

LD *

0 ' Fm-lm

,o

2.2

1

1

;

Az is random matrix of order 200. The right term is

A i x = bl

F=

.a.

,.

. .

'I1 F12

1

tively. Initial iteration value is zero.

I

and where Cii is orthogonal sub-matrix. i = 1,2, rn . So (4) is equivalent to .

A1 =

5/4 1 0

Divergence

I Convergence

1 Convergence

The iteration precision is l o p 6in table 1. Indicated from this table, algorithm 2 . 2 is stable and it can increase the rate of convergence. Because this decomposition is a method of elimination virtually, so it can be summed up to Eq. ( 2 ) , for this reason, the spectrum value be adjusted well, and the iteration be converged quickly. On the other hand, this study can improve the cognition to precondition. The program achieved in virtue of Mat labf7].

REFERANCES [ l ] Lu Tao, Shih T M, Liem C B. Domain decomposition m e t h e n e w numerical techniques for solving PDE. Beijing: Publishing House of Science, 1997. 102-291. [21 Hu Jiagan. Iteration solution for linear equations. Beijing : Publishing House o f Science, 1991. 105- 192. [3] Cai Dayong, ChenYmng. Solving power flow equations with inexact newton methods preconditioned by incomplete LU factorizationwith partially fill-in. Automation ofElectric Pozeter Systems. 2002,26(8): 11-14. [4] Luo Mngqiu, Liu Hong, Li Youming. LU decomposition

with spectral factorization in seismic imaging. Chinese Journal of &~%ysics. 2003,46(3): 421-427. [ 51 Wei Jianing, Wang Weicang, Pi xinming. A sort of multigrid parallel algorithms with virtual-boundary preconditionem. Journal o f Wuhan University of Technology ( Transportation Science & Engineering). 2001,25(1):4-7.

222 [6] Wei Jianing, Wang Zhongjun, Wang weicang. For-con&tion iteration characters and applications in grid equation parallel computation. Journal of Naual Uniuersity of Engineering. 2001, ( 5 ) . [7]Su Jinming, Ruan Shenyong. Application guide of MATLAB6.1. Beijing : Publishing house of Electronics Industry. 2002. 337-343.

W u Ruichan & Wei Jianing

Wu Ruichan was born in 1978. Now she is a candidate for M. S. degree of Wuhan University of Technology. Her main research areas are complexity theory and parallel computation.

Wei Jianing was born in 1946. Now he is a professor and advisor of master’ s candidate in Wuhan University of Technology. His main research areas are Approximation theory and parallel computation.