• 대한수학회보
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• 제37권3호
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• pp.551-568
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• 2000

We propose new parallelizable block incomplete factorization preconditioners for a symmetric block-tridiagonal H-matrix. Theoretical properties of these block preconditioners are compared with those of block incomplete factorization preconditioners for the corresponding comparison matrix. Numerical results of the preconditioned CG(PCG) method using these block preconditioners are compared with those of PCG method using a standard incomplete factorization preconditioner to see the effectiveness of the block incomplete factorization preconditioners.

• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.705-720
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• 2001

We propose a variant of parallel block incomplete factorization preconditioners for a symmetric block-tridiagonal H-matrix. Theoretical properties of these block preconditioners are compared with those of block incomplete factoriztion preconditioners for the corresponding somparison matrix. Numerical results of the preconditioned CG(PCG) method using these block preconditioners are compared with those of PCG using other types of block incomplete factorization preconditioners. Lastly, parallel computations of the block incomplete factorization preconditioners are carried out on the Cray C90.

• 대한기계학회논문집A
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• 제27권2호
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• pp.276-284
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• 2003

The preconditioned conjugate gradient method is an efficient iterative solution scheme for large size finite element problems. As preconditioning method, we choose an incomplete Cholesky factorization which has efficiency and easiness in implementation in this paper. The incomplete Cholesky factorization mettled sometimes leads to breakdown of the computational procedure that means pivots in the matrix become minus during factorization. So, it is inevitable that a reduction process fur stabilizing and this process will guarantee robustness of the algorithm at the cost of a little computation. Recently incomplete factorization that enhances robustness through increasing diagonal dominancy instead of reduction process has been developed. This method has better efficiency for the problem that has rotational degree of freedom but is sensitive to parameters and the breakdown can be occurred occasionally. Therefore, this paper presents new method that guarantees robustness for this method. Numerical experiment shows that the present method guarantees robustness without further efficiency loss.

• 대한수학회보
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• 제39권3호
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• pp.495-509
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• 2002

We propose variants of the modified incomplete Cho1esky factorization preconditioner for a symmetric positive definite (SPD) matrix. Spectral properties of these preconditioners are discussed, and then numerical results of the preconditioned CG (PCG) method using these preconditioners are provided to see the effectiveness of the preconditioners.

• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.59-80
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• 2003

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

• 대한수학회지
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• 제36권1호
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• pp.209-227
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• 1999

We propose new parallel block ILU (Incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix. Theoretial properties of these block preconditioners are studied to see the convergence rate of the preconditioned iterative methods, Lastly, numerical results of the right preconditioned GMRES and BiCGSTAB methods using the block ILU preconditioners are compared with those of these two iterative methods using a standard ILU preconditioner to see the effectiveness of the block ILU preconditioners.

• 한국국방경영분석학회지
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• 제24권1호
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• pp.146-156
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• 1998

Cholesky factorization is known to be inefficient to problems with dense column and network problems in interior point methods. We use the conjugate gradient method and preconditioners to improve the convergence rate of the conjugate gradient method. Several preconditioners were applied to LPABO 5.1 and the results were compared with those of CPLEX 3.0. The conjugate gradient method shows to be more efficient than Cholesky factorization to problems with dense columns and network problems. The incomplete Cholesky factorization preconditioner shows to be the most efficient among the preconditioners.

• Journal of applied mathematics & informatics
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• 제8권2호
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• pp.305-326
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• 2001

We introduce a class of multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This techniques is based on a recursive two by two block incomplete LU factorization on the coefficient martix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. Such a reduction process is repeated to yield a multilevel recursive preconditioner.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.307-316
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• 2006

Incomplete LU factorization preconditioning techniques often have difficulty on indefinite sparse matrices. We present hybrid reordering strategies to deal with such matrices, which include new diagonal reorderings that are in conjunction with a symmetric nondecreasing degree algorithm. We first use the diagonal reorderings to efficiently search for entries of single element rows and columns and/or the maximum absolute value to be placed on the diagonal for computing a nonsymmetric permutation. To augment the effectiveness of the diagonal reorderings, a nondecreasing degree algorithm is applied to reduce the amount of fill-in during the ILU factorization. With the reordered matrices, we achieve a noticeable improvement in enhancing the stability of incomplete LU factorizations. Consequently, we reduce the convergence cost of the preconditioned Krylov subspace methods on solving the reordered indefinite matrices.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.19-35
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• 2004

We develop in this paper some preconditioners for sparse non-symmetric M-matrices, which combine a recursive two-level block I LU factorization with multigrid method, we compare these preconditioners on matrices arising from discretized convection-diffusion equations using up-wind finite difference schemes and multigrid orderings, some comparison theorems and experiment results are demonstrated.