• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2000.11a
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• pp.484-488
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• 2000

병렬 컴퓨터를 사용하는 경우 하드웨어만으로 모든 것이 해결되지 않으며 병렬처리 기법의 도입이 불가피하다. 효과를 극대화하기 위하여서는 각 병렬 컴퓨터의 하드웨어적인 특징을 극대화할 수 있는 병렬 알고리즘과 병렬 프로그램 등 소프트웨어 개발이 필수적이다. GMRES(Generalized Minimal residual) 방법은 아주 큰 대칭 또는 비대칭 선형시스템의 해를 구하는 반복법 중의 하나로 일반적으로 많이 사용되고 있다. 서로 직교인 벡터를 하나씩 구하는 대신에 선형인 s개의 벡터를 구하고 각 그룹간에는 직교가 되게하는 s-step GMRES 알고리즘은 병렬적 성질을 더 많이 가지고 있다. 이 병렬 알고리즘의 전반부는 이미 개발된 s-step Arnoldi 알고리즘을 이용할 수 있다. s-step GMRES 알고리즘은 message passing 병렬 시스템에서 모든 프로세서들 사이의 자료 교환 시간을 줄임으로써 기존의 GMRES 방법에 비해 훨씬 더 병렬성을 증가시킨다. 본 논문에서는 초병렬 시스템(MPP)인 Cray T3E에서 많은 프로세서를 이용할 경우 개발된 s-step 알고리즘이 기존의 알고리즘에 비하여 얼마나 더 효과적으로 빨리 수행될 수 있는지 분석한다.

• 한국전산유체공학회:학술대회논문집
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• 2000.05a
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• pp.178-183
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• 2000

An efficient Newton-GMRES algorithm is presented for computing two-dimensional steady compressible viscous flows on unstructured hybrid meshes. The scheme is designed on cell-centered finite volume method which accepts general polygonal meshes. Steady-state solution is obtained with pseudo-transient continuation strategy. The preconditioned, restarted general minimum residual(GMRES) method is employed in matrix-free form to solve the linear system arising at each Newton iteration. The incomplete LU fartorization is employed for the preconditioning of linear system. The Spalart-Allmars one equation turbulence model is fully coupled with the flow equations to simulate turbulence effect. The accuracy, efficiency and robustness of the presently developed method are demonstrated on various test problems including laminar and turbulent flows over flat plate and airfoils.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.57-70
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• 1997

In this paper two preconditioned GMRES and QMR methods are applied to the non-Hermitian system from the Petrov-Galerkin procedure for the Poisson equation and compared to each other. To our purpose the ILUT and the SSOR preconditioners are used.

• 한국전산유체공학회:학술대회논문집
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• 1999.05a
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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.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.375-388
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• 2004

For solving symmetric systems of linear equations, it is shown that a new Krylov subspace method can be obtained. The new approach is one of the projection methods, and we call it the projection method for convenience in this paper. The projection method maintains the residual vector like simpler GMRES, symmetric QMR, SYMMLQ, and MINRES. By studying the quasiminimal residual method, we show that an extended projection method and the scaled symmetric QMR method are equivalent.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1131-1141
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• 2010

We develop an algorithm for computing a sparse approximate inverse for a nonsymmetric positive definite matrix based upon the FFAPINV algorithm. The sparse approximate inverse is computed in the factored form and used to work with some Krylov subspace methods. The preconditioner is breakdown free and, when used in conjunction with Krylov-subspace-based iterative solvers such as the GMRES algorithm, results in reliable solvers. Some numerical experiments are given to show the efficiency of the preconditioner.

• Nuclear Engineering and Technology
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• v.49 no.6
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• pp.1114-1124
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• 2017

Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates ($S_N$) or spherical-harmonics ($P_N$) solve to accelerate convergence of a high-order $S_N$ source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.3
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• pp.305-320
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• 2020

This study aims to generalize MINRES-N2 method [1]. It means that we tend to obtain an algorithm to transfer each normal matrix - that its eigenvalues belong to an algebraic curve of low degree k- to its condensed form through using a unitary similarity transformation. Then, we aim to obtain a method to solve a system of linear equations that its coefficient matrix is equal to such a matrix by utilizing it. Finally this method is compared to the well-known GMRES method through using numerical examples. The results obtained through examples show that the given method is more efficient than GMRES.

• East Asian mathematical journal
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• v.15 no.2
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• pp.345-354
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• 1999

Orthogonalization can be done by the well known Gram-Schmidt process or by using Householder transformations. In this paper, we introduced an alternative process using linear systems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.29-36
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• 1999

It is shown that a new Krylov subspace method for solving symmetric indefinite systems of linear equations can be obtained. We call the method as the projection method in this paper. The residual vector of the projection method is maintained at each iteration, which may be useful in some applications.