• 전기전자학회논문지
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• 제23권2호
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• pp.614-621
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• 2019

본 논문은 integral equation-fast Fourier transform(IE-FFT)과 block matrix preconditioner(BMP)를 이용하여 침투 가능한 구조물의 전자기 산란 문제를 다룬다. IE-FFT는 모멘트 법(the method of Moments : MoM)에 의해 형성된 행렬방정식의 해를 계산하기 위하여 반복법의 연산량을 상당히 개선할 수 있다. 또한 전기적으로 커다란 구조물로부터 형성된 행렬방정식에 BMP가 적용된 반복법을 적용하면 반복 횟수를 크게 줄여 행렬방정식의 해를 빠르게 계산할 수 있다. 수치해석 결과는 IE-FFT와 BMP를 적용하여 침투 가능한 구조물의 전자기 산란 문제를 빠르고 정확하게 계산할 수 있음을 보여준다.

• 대한기계학회논문집B
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• 제25권11호
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• pp.1581-1593
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• 2001

An efficient variable-reordering method for finite element meshes is used and the effect of variable-reordering is investigated. For the element renumbering of unstructured meshes, Cuthill-McKee ordering is adopted. The newsy reordered global matrix has a much narrower bandwidth than the original one, making the ILU preconditioner perform bolter. The effect of variable reordering on the convergence behaviour of saddle point type matrix it studied, which results from P2/P1 element discretization of the Navier-Stokes equations. We also propose and test 'level(0) preconditioner'and 'level(2) ILU preconditioner', which are another versions of the existing 'level(1) ILU preconditioner', for the global matrix generated by P2/P1 finite element method of incompressible Navier-Stokes equations. We show that 'level(2) ILU preconditioner'performs much better than the others only with a little extra computations.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.213-222
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• 2008

Iterative methods are often suitable for solving least squares problems min$||Ax-b||_2$, where A $\epsilon\;\mathbb{R}^{m{\times}n}$ is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the $A^T$ A-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix $A^T$ A. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min$||Ax-b||_2$. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner.

• 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 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.

• 대한수학회논문집
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• 제28권3호
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• pp.603-613
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• 2013

In 2009, Zheng and Miao [B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), 922-930] considered the modified Gauss-Seidel method for solving M-matrix linear system with the preconditioner $P_{max}$. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner $P_{max}({\alpha})$, and study its convergent properties when the coefficient matrix is an H-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.

• 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.

• 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.

• 디지털콘텐츠학회 논문지
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• 제10권4호
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• pp.579-585
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• 2009

Tyrtshnikov[9]의 연구에서는 토플리츠 선형시스템에서 토플리츠 선행조건으로 일반해를 구하는 방법들을 제시하고 있다. 또한 대칭 토플리츠 행렬에서의 선행조건 행렬을 선택하는 방법도 소개 하였다. 본 연구는 토플리츠 시스템에서 새롭게 선행조건 찾는 방법을 소개하고 있으며, 선행조건행렬들의 분석을 통해 대칭 토플리츠 행렬의 고유값들과 대칭 토플리츠행렬로 부터 생성된 선행조건행렬의 고유값들이 매우 근접하다는 결과를 나타내고 있다. 즉, 선행조건시스템 $C_0^{-1}T$의 고유값들은 1에 모두 접근하게되면, 선행조건 시스템의 수렴속도는 superlinear이다. 본 연구에서 생성된 선행조건행렬 $C_0$은 선행조건시스템의 superlinear의 수렴속도로 계산하게 된다. 또한 토플리츠 행렬은 이미지 프로세싱이나 시그널 프로세싱에서 많이 응용되고 있으므로 본 연구에서 개발한 선행조건행렬로부터 다양한 응용성을 높일 수 있다. 본연구의 또 다른 특징은 토플리츠 행렬의 중요한 성질을 보존하면서 선행조건행렬을 생성하였다.

• 대한수학회논문집
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• 제34권2호
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• pp.657-670
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• 2019

In this paper, we consider the preconditioned iterative methods for solving linear complementarity problem associated with an M-matrix. Based on the generalized Gunawardena's preconditioner, two preconditioned SSOR methods for solving the linear complementarity problem are proposed. The convergence of the proposed methods are analyzed, and the comparison results are derived. The comparison results showed that preconditioned SSOR methods accelerate the convergent rate of the original SSOR method. Numerical examples are used to illustrate the theoretical results.