• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.331-339
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• 2002

A preconditioned conjugate gradient(PCG) scheme with the aid of deflation for computing a few of the smallest eigenvalues arid their corresponding eigenvectors of the large generalized eigenproblems is considered. Topically there are two types of deflation techniques, the deflation with partial shifts and an arthogonal deflation. The efficient way of determining partial shifts is suggested and the deflation-PCG schemes with various partial shifts are investigated. Comparisons of theme schemes are made with orthogonal deflation-PCG, and their asymptotic behaviors with restart operation are also discussed.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제3권1호
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• pp.99-106
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• 1999

An accelerated optimization technique combined with a stepwise deflation procedure is presented for the efficient evaluation of a few of the smallest eigenvalues and their corresponding eigenvectors of the generalized eigenproblems. The optimization is performed on the Rayleigh quotient of the deflated matrices by the aid of a preconditioned conjugate gradient scheme with the incomplete Cholesky factorization.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.467-474
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• 2006

We present and study the stability and convergence of a deflation-preconditioned conjugate gradient(PCG) scheme for the interior generalized eigenvalue problem $Ax = {\lambda}Bx$, where A and B are large sparse symmetric positive definite matrices. Numerical experiments are also presented to support our theoretical results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권2호
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• pp.17-23
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• 2001

The evaluation of a few of the smallest eigenpairs of large symmetric eigenvalue problem is of great interest in many physical and engineering applications. A deflation-preconditioned conjugate gradient(PCG) scheme for a such problem has been shown to be very efficient. In the present paper we provide the numerical stability of a deflation-PCG with partial shifts.

• 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.

• 대한기계학회논문집A
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• 제24권8호
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• pp.2100-2107
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• 2000

The wavelet theory is relatively a new development and now acquires popularity and much interest in many areas including mathematics and engineering. This work presents an adaptive wavelet method for a numerical solution of partial differential equations in a collocation sense. Due to the multi-resolution nature of wavelets, an adaptive strategy can be easily realized it is easy to add or delete the wavelet coefficients as resolution levels progress. Typical wavelet-collocation methods use interpolating wavelets having no vanishing moment, but we propose a new wavelet-collocation method on modified interpolating wavelets having 2 vanishing moments. The use of the modified interpolating wavelets obtained by the lifting scheme requires a smaller number of wavelet coefficients as well as a smaller condition number of system matrices. The latter property makes a preconditioned conjugate gradient solver more useful for efficient analysis.

• 한국전산유체공학회지
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• 제19권4호
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• pp.20-28
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• 2014

An efficient solution algorithm for simulating free surface problem is presented. Navier-Stokes equations for variable density incompressible flow are employed as the governing equation on Cartesian meshes. In order to describe the free surface motion efficiently, VOF(Volume Of Fluid) method utilizing THINC(Tangent of Hyperbola for Interface Capturing) scheme is employed. The most time-consuming part of the current free surface flow simulations is the solution step of the linear system, derived by the pressure Poisson equation. To solve a pressure Poisson equation efficiently, the PCG(Preconditioned Conjugate Gradient) method is utilized. This study showed that the proper application of the preconditioner is the key for the efficient solution of the free surface flow when its pressure Poisson equation is solved by the CG method. To demonstrate the efficiency of the current approach, we compared the convergence histories of different algorithms for solving the pressure Poisson equation.

• 대한토목학회논문집
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• 제13권3호
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• pp.119-127
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• 1993

근래 천수방정식과 같은 2차원 수치해석에서 가장 널리 쓰이는 방법 중의 하나로 ADI 방법을 들 수 있다. 그러나 서해안과 같이 수심의 변화가 심하며 특히 해저협곡이 곳곳에 발달된 해역에서 조석에 관한 문제해결시 ADI 방법을 사용하면 소위 ADI 효과가 크게 우려된다. 이를 극복하기 위하여 완전 음해법으로 차분되고 CGS(conjugate gradient squared) 방법으로 해를 구하는 알고리즘을 개발하였다. 조간대 모의가 포함된 본 모형용 새만금 수역에 적용한 결과 지형의 복잡성에도 불구하고 유속장과 조간대 형성에 관한 수치적 모의가 만족스러운 결과를 보였다.

• 한국전산구조공학회논문집
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• 제34권3호
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• pp.151-157
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• 2021

본 연구에서는 비국부 적분 연산기로 표현되는 페리다이나믹 방정식의 수렴성을 검토한다. 정적/준정적 손상 해석 문제를 효율적으로 해석하기 위해 페리다이나믹 방정식의 implicit 정식화가 필요하다. 이 과정에서 페리다이나믹 비국부 적분 방정식으로부터 대수방정식 형태가 나타나게 되어 시스템 행렬 계산을 위해 많은 시간이 소요되기 때문에, 효율적인 계산을 위해 수렴성이 중요한 요소가 된다. 특히 radial influence 함수를 적분 kernel로 사용하는 경우 fractional Laplacian 적분 방정식이 유도된다. 비국부 적분 연산기의 교윳값 성질에 의해 대수방정식의 condition number가 radial influence 함수의 차수 및 비국부 영역의 크기에 영향을 받는 것이 수학적으로 확인되었다. 본 연구에서는 이를 토대로 균열이 있는 페리다이나믹 정적 해석 문제를 Newton-Raphson 방법으로 해석할 때 적분 커널의 차수, 비국부 영역의 크기 등이 대수방정식의 condition number와 preconditioned conjugate gradient (PCG) 방법으로 계산 시 수렴성 및 계산 시간에 미치는 영향을 수치적으로 분석한다.

• Nuclear Engineering and Technology
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• 제52권3호
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• pp.485-498
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• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.