• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.305-316
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• 2003

Recently, an iterative algorithm for finding the interior eigenvalues of a definite matrix by CG-type method has been proposed. This method compares to the inverse power method. The given matrices A, and B are assumed to be large and sparse, and SPD( Symmetric Positive Definite) The CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for smallest eigenvalue. Also, it is very amenable to parallel computations, like the CG method for the linear systems. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. But for parallel computations we need to find an efficient parallel preconditioner. Our candidates we ILU(0) in the wave-front order, ILU(0) in the multi-coloring order, Point-SSOR(Symmetric Successive Overrelaxation), and Multi-Color Block SSOR preconditioner. Wavefront order is a simple way to increase parallelism in the natural order, and Multi-coloring realizes a parallelism of order(N), where N is the order of the matrix. Another choice is the Multi-Color Block SSOR(Symmetric Successive OverRelaxation) preconditioning. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test problem was drawn from the discretizations of partial differential equations by finite difference methods. The results show that for small number of processors Multi-Color ILU(0) has the best performance, while for large number of processors Multi-Color Block SSOR performs the best.

• 정보처리학회논문지A
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• 제10A권2호
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• pp.137-140
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• 2003

최근에 CG 반복법을 이용하여 레이레이 계수를 최소화함으로써 대칭행렬의 내부고유치를 구하는 방법이 개발되었다 그리고 이 방법은 병렬계산에 매우 적합하다. 적절한 준비행렬의 선택은 수렴속도를 향상시킨다. 우리는 본 연구에서 이를 위한 병렬준비행렬들을 비교한다. 고려된 준비행렬들은 Point-SSOR, 다중색채하의 ILU(0)와 Block SSOR이다. 우리는 128개의 노드를 가진 CRAY-T3E에서 구현하였다. 프로세서간의 통신은 MPI 리이브러리를 사용하였다. 최고 512$\times$512 행렬까지 시험하였는데 이 행렬들은 타원형 편미분방정식의 근사화에서 얻어졌다. 그 결과 다중색채 Block SSOR이 가장 성능이 우수한 것으로 판명되었다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권2호
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• pp.63-69
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• 2001

In this study, we shall be concerned with computing in parallel a few of the smallest eigenvalues and their corresponding eigenvectors of the eigenvalue problem, $Ax={\lambda}Bx$, where A is symmetric, and B is symmetric positive definite. Both A and B are large and sparse. Recently iterative algorithms based on the optimization of the Rayleigh quotient have been developed, and CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for small extreme eigenvalues. As in the case of a system of linear equations, successful application of the CG scheme to eigenproblems depends also upon the preconditioning techniques. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. The idea underlying the present work is a parallel computation of the Multi-Color Block SSOR preconditioning for the CG optimization of the Rayleigh quotient together with deflation techniques. Multi-Coloring is a simple technique to obatin the parallelism of order n, where n is the dimension of the matrix. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI(Message Passing Interface) library was adopted for the interprocessor communications. The test problems were drawn from the discretizations of partial differential equations by finite difference methods.

• 정보처리학회논문지A
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• 제8A권3호
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• pp.227-234
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• 2001

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditiner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to $1025{\times}1024$. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications, The results show that Multi-Color Block SOR is scalabl and gives the best performances.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권1호
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• pp.85-100
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• 2001

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditioner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024 x 1024, and for an ill-conditioned matrix from the shell problem from the Harwell-Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR converges.