• Title, Summary, Keyword: preconditioned conjugate gradient scheme

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AN ACCELERATED DEFLATION TECHNIQUE FOR LARGE SYMMETRIC GENERALIZED EIGENPROBLEMS

  • HYON, YUN-KYONG;JANG, HO-JONG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.3 no.1
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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.

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PERTURBATION ANALYSIS OF DEFLATION TECHNIQUE FOR SYMMETRIC EIGENVALUE PROBLEM

  • JANG, HO-JONG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.2
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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.

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An Efficient Adaptive Wavelet-Collocation Method Using Lifted Interpolating Wavelets (수정된 보간 웨이블렛응 이용한 적응 웨이블렛-콜로케이션 기법)

  • Kim, Yun-Yeong;Kim, Jae-Eun
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.24 no.8
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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.

Study on Robustness of Incomplete Cholesky Factorization using Preconditioning for Conjugate Gradient Method (불완전분해법을 전처리로 하는 공액구배법의 안정화에 대한 연구)

  • Ko, Jin-Hwan;Lee, Byung-Chai
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.27 no.2
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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.

AN EFFICIENT ALGORITHM FOR INCOMPRESSIBLE FREE SURFACE FLOW ON CARTESIAN MESHES (직교격자상에서 효율적인 비압축성 자유표면유동 해법)

  • Go, G.S.;Ahn, H.T.
    • Journal of computational fluids engineering
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    • v.19 no.4
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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.

Numerical Analysis of Shallow Water Equation with Fully Implicit Method (음해법을 이용한 천수방정식의 수치해석)

  • Kang, Ju Whan;Park, Sang Hyun;Lee, Kil Seong
    • Journal of The Korean Society of Civil Engineers
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    • v.13 no.3
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    • pp.119-127
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    • 1993
  • Recently, ADI scheme has been a most common tool for solving shallow water equation numerically. But ADI models of tidal flow is likely to cause so called ADI effect in such a region of the Yellow Sea which shows complex topography and has submarine canyons especially. To overcome this, a finite difference algorithm is developed which adopts fully implicit method and preconditioned conjugate gradient squared method. Applying the algorithm including simulation of intertidal zone to Sae-Man-Keum. velocity fields and flooding/drying phenomena are simulated well in spite of complex topography.

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Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for SN transport calculations

  • Muhammad, Habib;Hong, Ser Gi
    • Nuclear Engineering and Technology
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    • v.52 no.3
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    • pp.485-498
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    • 2020
  • A diffusion synthetic acceleration (DSA) technique for the SN 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.

Finite Element Analysis of Shape Rolling Process using Destributive Parallel Algorithms on Cray T3E (병렬 컴퓨터를 이용한 형상 압연공정 유한요소 해석의 분산병렬처리에 관한 연구)

  • Gwon, Gi-Chan;Yun, Seong-Gi
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.24 no.5
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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.

Comparison of PCGM and Parabolic Approximation Numerical Models for an Elliptic Shoal (타원형천퇴에 대한 PCGM과 포물형근사식 수치모형비교)

  • 서승남;연영진
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.6 no.3
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    • pp.216-225
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    • 1994
  • By use of laboratory experiment data set for an elliptic shoal by Berkhoff et al. (1982), both accuracy and Performance tests of numerical results between PCGM (Preconditioned Conjugate Gradient Method) and PA(Parabolic Approximation) are compared. Although both results show good agreement with the experimental data the PA model gives better reproduction of the relatively high amplitudes in the section 4-5 downwave of the shoal, in comparison with the PCGM. The PA model has been proved to be a useful tool for predicting wave transformationsin large shallow water region, but it can be applied only to the case of negligible reflection. On the other hand, there is a need to improve the computational efficiency of the PCGM model which is a finite difference scheme directly derived from the mild slope equation and can handle reflection. By taking the results of th PA model as an input data of the PCGM, the CPU time can be reduced by about 40%.

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