• Transactions of the Korean Society of Mechanical Engineers A
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• v.21 no.11
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• pp.1852-1861
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• 1997

An enhanced subspace iteration algorithm has been developed to solve eigenvalue problems reliably and efficiently. Basic subspace iteration algorithm has been improved by eliminating recalculation of converged eigenvectors, using Krylov sequence as initial vectors and incorporating with shifting techniques. The number of iterations and computational time have been considerably reduced when compared with the original one, and reliability for catching copies of the multiple roots has been retained successfully. Further research would be required for mathematical justification of the present method.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.23 no.2
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• pp.153-163
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• 2010

In this paper a frequency response analysis using Krylov subspace-based model reduction and its design sensitivity analysis with respect to design variables are presented. Since the frequency response and its design sensitivity information are necessary for a gradient-based optimization, problems of high computational cost and resource may occur in the case that frequency response of a large sized finite element model is involved in the optimization iterations. In the suggested method model order reduction of finite element models are used to calculate both frequency response and frequency response sensitivity, therefore one can maximize the speed of numerical computation for the frequency response and its design sensitivity. As numerical examples, a semi-monocoque shell and an array-type $4{\times}4$ MEMS resonator are adopted to show the accuracy and efficiency of the suggested approach in calculating the FRF and its design sensitivity. The frequency response sensitivity through the model reduction shows a great time reduction in numerical computation and a good agreement with that from the initial full finite element model.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.1-4
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• 1999

The Helmholtz equation is very important in physics and engineering. However, solution of the Helmholtz equation is in general known as a very difficult phenomenon. For if the ${\omega}$ is negative, the FDM discretized linear system becomes indefinite, whose solution by iterative method requires a very clever preconditioner. In this paper we assume that ${\omega}$ is nonnegative, and determine the optimal ${\rho}$ parameter for the three dimensional ADI iteration for the Helmholtz equation. The ADI(Alternating Direction Implicit) method is also getting new attentions due to the fact that it is very suitable to the vector/parallel computers, for example, as a preconditioner to the Krylov subspace methods. However, classical ADI was developed for two dimensions, and for three dimensions it is known that its convergence behaviour is quite different from that in two dimensions. So far, in three dimensions the so-called Douglas-Rachford form of ADI was developed. It is known to converge for a relatively wide range of ${\rho}$ values but its convergence is very slow. In this paper we determine the necessary conditions of the ${\rho}$ parameter for the convergence and optimal ${\rho}$ for the three dimensional ADI iteration of the Peaceman-Rachford form for the real Helmholtz equation with nonnegative ${\omega}$. Also, we conducted some experiments which is in close agreement with our theory. This straightforward extension of Peaceman-rachford ADI into three dimensions will be useful as an iterative solver itself or as a preconditioner to the the Krylov subspace methods, such as CG(Conjugate Gradient) method or GMRES(m).

• Transactions of the Korean Society of Mechanical Engineers A
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• v.31 no.5
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• pp.541-549
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• 2007

In the case of control for mechanical systems, it is highly useful to be able to provide a compact model of the mechanical system to control engineers using the smallest number of state variables, while still providing an accurate model. The reduced mechanical model can then be inserted into the complete system models and used for extended system-level dynamic simulation. In this paper, moment-matching based model order reductions (MOR) using Krylov subspaces, which reduce the number of degrees of freedom of an original finite element model via the Arnoldi process, are presented to study the eigenvalue and frequency response problems of a HDD actuator and suspension system.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.4
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• pp.375-385
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• 2020

In this paper, a special indefinite inner product, named hyperbolic scalar product, is used and all acquired results have been raised and proved with the proviso that the space is equipped with this indefinite scalar product. The main objective is to be introduced and applied an indefinite oblique projection method, called Indefinite Lanczos J-biorthogonalizatiom process, which in addition to building a pair of J-biorthogonal bases for two used Krylov subspaces, leads to the introduction of a process for solving large non-J-symmetric linear systems, i.e., Indefinite two-sided Lanczos Algorithm for 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.5-16
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• 1999

We introduce a new projector for accelerating convergence of a symmetric eigenvalue problem Ax = x, and devise a power/Lanczos hybrid algorithm. Acceleration can be achieved by removing the hard-to-annihilate nonsolution eigencomponents corresponding to the widespread eigenvalues with modulus close to 1, by estimating them accurately using the Lanczos method. However, the additional Lanczos results can be obtained without expensive matrix-vector multiplications but a very small amount of extra work, by utilizing simple power-Lanczos interconversion algorithms suggested. Numerical experiments are given at the end.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.145-155
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• 1997

We present a couple of tools that can be used in the solution of nonsymmetric eigenvalue problems. The first one allows us to convert power iterates into Arnoldi's results so that a few eigenpairs are easily obtainable. The other converts Arnoldi's results into power iterates to simulate the power method and improve the result. Suggestions for application are also given.

• Journal of applied mathematics & informatics
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• v.8 no.2
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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.

• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.667-678
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• 1997

This paper contains a convergence analysis of a flexible restarted FOM(k)(FFOM(k)), and its performance is compared with FGMRES(k). Performances of these two algorithms with variable preconditioners are also compared with those of preconditioned FOM(k) and GMRES(k). Numerical experiments show that FFOM(k) performs as well as, or better than for some problems, FGMRES(k).

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.155-162
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• 2018

We describe a parallel implementation of a relaxed Hermitian and skew-Hermitian splitting preconditioner for the numerical solution of saddle point problems arising from the steady incompressible Navier-Stokes equations. The equations are linearized by the Picard iteration and discretized with the finite element and finite difference schemes on two-dimensional and three-dimensional domains. We report strong scalability results for up to 32 cores.