• 대한기계학회논문집A
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• 제24권5호
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• pp.1215-1230
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• 2000

Parallel Approaches using Cray T3E which is NIPP (Massively Parallel Processors) machine are presented for the efficient computation of the finite element analysis of 3-D shape rolling processes. D omain decomposition method coupled with parallel linear equation solver is used. Domain decomposition is applied for obtaining element tangent stifffiess matrices and residual vectors. Direct and iterative parallel algorithms are used for solving the linear equations. Direct algorithm is_parallel version of direct banded matrix solver. For iterative algorithms, the well-known preconditioned conjugate gradient solver with Jacobi preconditioner is also employed. Moreover a new effective iterative scheme with block inverse matrix preconditioner, which is named by present authors, is presented and its results are compared with the one using Jacobi preconditioner. PVM and MPI are used for message passing and synchronization between processors. The performance and efficiency of each algorithm is discussed and comparisons are made among different algorithms.

• Wind and Structures
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• 제22권3호
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• pp.349-368
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• 2016

A preconditioning technique is presented for a simultaneous solution to wind-membrane interaction. In the simultaneous equations, a linear elastic model was employed to deal with the fluid-structure data transfer at the interface. A Lagrange multiplier was introduced to impose the specified boundary conditions at the interface and strongly coupled simultaneous equations are derived after space and time discretization. An initial linear elastic model preconditioner and modified one were derived by treating the linearized elastic model equation as a saddle point problem, respectively. Accordingly, initial and modified fluid-structure interaction (FSI) preconditioner for the simultaneous equations were derived based on the initial and modified linear elastic model preconditioners, respectively. Wind-membrane interaction analysis by the proposed preconditioners, for two and three dimensional membranous structures respectively, was performed. Comparison was made between the performance of initial and modified preconditioners by comparing parameters such as iteration numbers, relative residuals and convergence in FSI computation. The results show that the proposed preconditioning technique greatly improves calculation accuracy and efficiency. The priority of the modified FSI preconditioner is verified. The proposed preconditioning technique provides an efficient solution procedure and paves the way for practical application of simultaneous solution for wind-structure interaction computation.

• 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.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.283-296
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• 2005

It is well known that the ordering of the unknowns can have a significant effect on the convergence of a preconditioned iterative method and on its implementation on a parallel computer. To do so, we introduce a block red-black coloring to increase the degree of parallelism in the application of the block ILU preconditioner for solving sparse matrices, arising from convection-diffusion equations discretized using the finite difference scheme (five-point operator). We study the preconditioned PGMRES iterative method for solving these linear systems.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.351-361
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• 2010

The solution of large sparse linear systems is one of the most important problems in large scale scientific computing. Among the many methods developed, the preconditioned Krylov subspace methods are considered the preferred methods. Selecting a suitable preconditioner with appropriate parameters for a specific sparse linear system presents a challenging task for many application scientists and engineers who have little knowledge of preconditioned iterative methods. The prediction of ILU type preconditioners was considered in [27] where support vector machine(SVM), as a data mining technique, is used to classify large sparse linear systems and predict best preconditioners. In this paper, we apply the data mining approach to the sparse approximate inverse(SAI) type preconditioners to find some parameters with which the preconditioned Krylov subspace method on the linear systems shows best performance.

• 한국항공우주학회지
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• 제35권7호
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• pp.586-591
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• 2007

예조건화 오일러 방정식의 수렴특성에 미치는 계산 오차의 영향을 해석하였다. 다양한 마하수의 원형 턱이 있는 이차원 관내 유동을 계산하였다. 마하수가 감소함에 따라 차분오차는 증가하는데, 에너지 방정식의 계산 오차는 다른 방정식의 계산 오차보다 빠르게 증가하는 것으로 나타났다. 그리고 예조건 행렬의 역행렬을 곱하여 변형된 지배방정식 형태를 이용하면 계산 오차 문제를 완화할 수 있음을 보였다

• 대한수학회지
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• 제57권5호
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• pp.1267-1286
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• 2020

We pay attention to a coupled problem by a penalty term which is induced from non-overlapping domain decomposition methods based on augmented Lagrangian methodology. The coupled problem is composed by two parts mainly; one is a problem associated with local problems in non-overlapping subdomains and the other is a coupled part over all subdomains due to the penalty term. For the speedup of iterative solvers for the coupled problem, we propose two different types of preconditioners: a block-diagonal preconditioner and an additive Schwarz preconditioner as overlapping domain decomposition methods. We analyze the coupled problem and the preconditioned problems in terms of their condition numbers. Finally we present numerical results which show the performance of the proposed methods.

• Journal of applied mathematics & informatics
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• 제36권3_4호
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• pp.317-330
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• 2018

We first provide how to apply the global preconditioned conjugate gradient ($G{\ell}-PCG$) method with Kronecker product preconditioners to image deblurring problems with nearly separable point spread functions. We next provide a coarse-grained parallel image deblurring algorithm using the $G{\ell}-PCG$. Lastly, we provide numerical experiments for image deblurring problems to evaluate the effectiveness of the $G{\ell}-PCG$ with Kronecker product preconditioner by comparing its performance with those of the $G{\ell}-CG$, CGLS and preconditioned CGLS (PCGLS) methods.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.687-696
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• 2010

In this paper, an algorithm for computing the sparse approximate inverse factor of matrix $A^{T}\;A$, where A is an $m\;{\times}\;n$ matrix with $m\;{\geq}\;n$ and rank(A) = n, is proposed. The computation of the inverse factor are done without computing the matrix $A^{T}\;A$. The computed sparse approximate inverse factor is applied as a preconditioner for solving normal equations in conjunction with the CGNR algorithm. Some numerical experiments on test matrices are presented to show the efficiency of the method. A comparison with some available methods is also included.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 1999년도 춘계 학술대회논문집
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• pp.106-112
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• 1999

In this article, combination of the FAS-FMG multi-grid method and the Krylov subspace method was presented in solving two dimensional driven-cavity flows. Three algorithms of the Krylov subspace method, CG, CGSTAB(Bi-CG Stabilized) and GMRES method were tested with MILU preconditioner. As a smoother of the pressure correction equation, the MILU-CG is recommended rather than MILU-GMRES(k) or MILU-CGSTAB, since the MILU-GMRES(k) preconditioner has too much computation on the coarse grid compared to the MILU-CG one. As for the momentum equation, relatively cheap smoother like SIP solver may be sufficient.