• 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.

• 호남수학학술지
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• 제32권3호
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• pp.399-412
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• 2010

In this paper, we consider a preconditioned accelerated overrelaxation (PAOR) method to solve systems of linear equations. We show the convergence of the PAOR method. We also give com-parison results when the coefficient matrix is an L- or H-matrix. Finally, we provide some numerical experiments to show efficiency of PAOR method.

• 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.

• 한국전자파학회논문지
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• 제9권6호
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• pp.735-745
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• 1998

모멘트법은 전자파산란문제에 널리 사용되고 있는데, 최근에 대용량의 문제를 빠르고 효율적으로 풀 수 있는 기법들에 대한 연구가 많이 진행되고 있다. 대부분의 이런 기법에는 계산속도나 기억용량을 효율적으로 이용할 수 있는 반복법을 사용해서 행렬방정식을 풀게 되는데, 유사공진특성을 갖는 물체에 대한 산란은 물체 내부에서 전자파가 공진하는 특성을 가지므로 반복해법올 이용하여 적분방정식을 풀 경우 수렴이 잘 되지 않거나, 수렴되기까지 많은 반복회수를 필요로 한다. 본 논문에서 사용된 MLFMA(Muli-level Fast Multipole Algorithm)는 FMM(Fast Multipole Method)을 다층으로 확장한 알고리듬으로 반복회수당 계산시간을 O(NlogN)으로 줄일 수 있다. 이 MLFMA를 유사공진형구조에 적용하고, 또한 행렬식을 블록밴드행렬 전처리를 하여 반복회수를 감소시켰다. 여기서 사용된 전처리행렬은 행렬분할법을 이용하여 O(N)의 계산시간으로 구할 수 있으므로, 미지수가 많을 때는 전처리행렬을 구하는데 드는 추가계산시간을 무시할 수 있다. 여기서 제안된 방법을 비행기의 공기유입구에 대한 TM전자파산란 계산에 적용하여 효율성을 보였다

• 한국전자파학회논문지
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• 제23권2호
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• pp.169-176
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• 2012

본 논문에서는 전계 적분 방정식 (Electric Field Integral Equation: EFIE)을 사용하는 모멘트 법의 저주파 오차(low frequency breakdown) 문제를 해결하기 위한 방법으로 루프-스타(loop-star) 기저 함수를 사용하였다. 또한, 모멘트 법의 해를 계산하기 위하여 conjugate gradient method(CGM)과 같은 반복법을 적용할 경우 반복 횟수를 줄이기 위한 기법으로 p-Type Multiplicative Schwarz preconditioner(pMUS)를 이용하였다. 헬름홀쯔 정리(Helmholtz theorem)에 기반한 루프-스타(loop-star) 기저 함수와 주파수 정규화 기법을 이용하여 전계 적분 방정식에서 Rao-Wilton-Glisson(RWG) 기저 함수를 사용하였을 때 발생하는 저주파 오차(low frequency instability) 문제를 해결할 수 있다. 하지만, RWG 기저 함수를 비발산(solenoidal) 성분과 비회전성(irroatational) 성분으로 분해함으로써 발생하는 행렬 방정식의 높은 조건 수(condition number)로 인하여 CGM과 같은 반복법을 사용할 경우 해를 계산하기 위하여 많은 반복 횟수가 요구된다. 본 논문에서는 이러한 문제점을 해결하기 위한 방안으로 pMUS 전제 조건 기법을 이용하여 CGM의 반복 횟수를 줄였다. 수치 해석 결과, pMUS와 같은 희소성(sparsity)을 가진 블럭 대각 전제 조건(Block Diagonal Precondtioner: BDP)과 비교하였을 때 pMUS는 BDP보다 빠르게 해를 계산할 수 있다.

• 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.

• 대한수학회논문집
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• 제11권3호
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• pp.867-880
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• 1996

Numerical approximations to the linear elasticity are traditionally based on the finite element method. In this paper we propose a new formulation based on the cubic spline collocation method for linear elastic problem on the unit square. We present several numerical results for the eigenvalues of the matrix represented by cubic collocation method and preconditioner matrix which is preconditioned by FEM and FDM. Finally we present the numerical solution for some example equation.

• 대한수학회보
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• 제38권1호
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• pp.17-27
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• 2001

We discuss a finite difference preconditioner for the$C^1$ Lagrance quadratic spline collocation method for a uniformly elliptic operator with homogeneous Dirichlet boundary conditions. Using the generalized field of values argument, we analyzed eigenvalues of the matrix preconditioned by the matrix corresponding to a finite difference operator with zero boundary condition.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 1998년도 추계 학술대회논문집
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• pp.153-159
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• 1998

The Newton-Krylov method on the unstructured grid flow solver using the cell-centered spatial discretization oi compressible Euler equations is presented. This flow solver uses the reconstructed primitive variables to get the higher order solutions. To get the quadratic convergence of Newton method with this solver, the careful linearization of face flux is performed with the reconstructed flow variables. The GMRES method is used to solve large sparse matrix and to improve the performance ILU preconditioner is adopted and vectorized with level scheduling algorithm. To get the quadratic convergence with the higher order schemes and to reduce the memory storage. the matrix-free implementation and Barth's matrix-vector method are implemented and compared with the traditional matrix-vector method. The convergence and computing times are compared with each other.

• Journal of applied mathematics & informatics
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• 제32권3_4호
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• pp.389-403
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• 2014

In this paper, we first provide comparison results of preconditioned AOR methods with direct preconditioners $I+{\beta}L$, $I+{\beta}U$ and $I+{\beta}(L+U)$ for solving a linear system whose coefficient matrix is a large sparse irreducible L-matrix, where ${\beta}$ > 0. Next we propose how to find a near optimal parameter ${\beta}$ for which Krylov subspace method with these direct preconditioners performs nearly best. Lastly numerical experiments are provided to compare the performance of preconditioned iterative methods and to illustrate the theoretical results.