• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제22권1호
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• pp.15-28
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• 2018

Recently Machine Learning algorithms are widely used to process Big Data in various applications and a lot of these applications are executed in run time. Therefore the speed of Machine Learning algorithms is a critical issue in these applications. However the most of modern iteration Machine Learning algorithms use a successive iteration technique well-known in Numerical Linear Algebra. But this technique has a very low convergence, needs a lot of iterations to get solution of considering problems and therefore a lot of time for processing even on modern multi-core computers and clusters. Tchebychev iteration technique is well-known in Numerical Linear Algebra as an attractive candidate to decrease the number of iterations in Machine Learning iteration algorithms and also to decrease the running time of these algorithms those is very important especially in run time applications. In this paper we consider the usage of Tchebychev iterations for acceleration of well-known K-Means and SVM (Support Vector Machine) clustering algorithms in Machine Leaning. Some examples of usage of our approach on modern multi-core computers under Apache Spark framework will be considered and discussed.

• Steel and Composite Structures
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• 제14권6호
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• pp.605-620
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• 2013

This paper provides an innovative iteration technique for the large deflection problem of annular plate. After some manipulation, the problem is reduced to a couple of ODEs (ordinary differential equation). Among them, one is derived from the plane stress problem for plate, and other is derived from the bending of plate. Since the large deflection for plate is assumed in the problem, the relevant non-linear terms appear in the resulting ODEs. The pseudo-linearization procedure is suggested to solve the problem and the nonlinear ODEs can be solved in the way for the solution of linear ODE. To obtain the final solution, it is necessary to use the iteration. Several numerical examples are provided. In the study, the assumed value for non-dimensional loading is larger than those in the available references.

• 한국방사선학회논문지
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• 제17권1호
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• pp.141-148
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• 2023

CT에서 대수적재구성기법은 잡음에 강점을 가지는 재구성기법으로 알려져 있다. 대수적재구성기법에서 반복 횟수는 계산 시간을 결정하는 중요한 인자 중의 하나이다. 그러나 반복 횟수에 대한 기준이 연구되어 있지만 수백 번 이상의 반복을 수행하게 되어 현실적으로 사용하기에는 무리가 있었다. 본 연구에서는 반복 횟수를 결정할 수 있는 현실적인 방법을 제시하였다. 반복 횟수에 따라 단면 영상의 품질이 천천히 개선된다는 것을 이용하였다. 반복 횟수를 절대치 평균 오차의 차이 𝜖 < 0.001 로 선택하였다. 잡음이 없는 경우는 Shepp-Logan 두부 팬텀을 이용하였고 잡음이 있는 경우는 Geant4를 이용하여 다양한 입사광자에 대해 360, 720, 1,440개의 투영을 얻었다. 정지 조건에서 10회 내외의 반복으로 우수한 단면 영상을 획득하였다. 수 백회 이상을 반복하는 최적 영상기반의 방법에 비해 현실적으로 적용 가능성이 높은 방법이 될 수 있을 것이다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권1호
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• pp.29-34
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• 2010

The QR method is one of the most common methods for calculating the eigenvalues of a square matrix, however its understanding would require complicated and sophisticated mathematical logics. In this article, we present a simple way to understand QR method only with a minimal mathematical knowledge. A deflation technique is introduced, and its combination with the power iteration leads to extracting all the eigenvectors. The orthogonal iteration is then shown to be compatible with the combination of deflation and power iteration. The connection of QR method to orthogonal iteration is then briefly reviewed. Our presentation is unique and easy to understand among many accounts for the QR method by introducing the orthogonal iteration in terms of deflation and power iteration.

• 한국지진공학회:학술대회논문집
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• 한국지진공학회 2003년도 춘계 학술발표회논문집
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• pp.481-490
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• 2003

This paper presents an efficient eigensolution method for non-proportionally damped systems. The proposed method is obtained by applying the accelerated Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linearized form of the quadratic eigenproblem. A step length and a selective scheme are introduced to increase the convergence of the solution. The step length can be evaluated by minimizing the norm of the residual vector using the least square method. While the singularity may occur during factorizing process in other iteration methods such as the inverse iteration method and the subspace iteration method if the shift value is close to an exact eigenvalue, the proposed method guarantees the nonsingularity by introducing the orthonormal condition of the eigenvectors, which can be proved analytically. A numerical example is presented to demonstrate the effectiveness of the proposed method.

• 한국경영과학회:학술대회논문집
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• 한국경영과학회 1997년도 추계학술대회발표논문집; 홍익대학교, 서울; 1 Nov. 1997
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• pp.52-55
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• 1997

In this paper, both the generalization of one artificial variable technique to the general bound problem and the efficient implementation of the technique are suggested. When the steepest-edge method is used as a pricing rule in the simplex method, it is easy to update the reduced cost and the simplex multiplier every iteration. Therefore, one artificial variable technique is more efficient than Wolfe's method in which the reduced cost and simplex multiplier must be recalculated in every iteration. When implementing the one artificial variable technique on the LP problems with the general bound restraints on the variables, an arbitrary basic solution which satisfies the bound restraints is sought first, and the artificial column which adjusts the infeasibility is introduced. The phase one of the simplex method minimizes the one artificial variable. The efficient implementation technique includes the splitting, scaling, storage of the artificial column, and the cure of infeasibility problem.

• 대한전자공학회논문지
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• 제21권1호
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• pp.25-31
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• 1984

본 논문에서는 시간제한신호에 있어서 종래의 반복외삽법의 전 과정을 단일한 연산으로 나타낼 수 있는 외삽행렬을 이용한 신호재생방법에 관한 알고리즘을 고안하고 또한 컴퓨터를 이용한 모의실험을 통하여 제안된 알고리즘을 초음파 진단장치에 적용, 정확성과 고속적인 측면에서 그 효용성을 입증하였다.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 1998년도 가을 학술발표회 논문집
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• pp.84-91
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• 1998

A numerically stable technique to remove tile limitation in choosing a shift in the subspace iteration method with shift is presented. A major difficulty of the subspace iteration method with shift is that because of singularity problem, a shift close to an eigenvalue can not be used, resulting in slower convergence. This study selves the above singularity problem using side conditions without sacrifice of convergence. The method is always nonsingular even if a shiht is an eigenvalue itself. This is one of tile significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, two numerical examples are considered

• Nonlinear Functional Analysis and Applications
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• 제27권1호
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• pp.59-82
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• 2022

In this paper, we present a modified (improved) generalized M-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized M-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

• Interaction and multiscale mechanics
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• 제1권2호
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• pp.211-230
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• 2008

Owing to the growing size of the eigenvalue problem and the growing number of eigenvalues desired, solution methods of iterative nature are becoming more popular than ever, which however suffer from low efficiency and lack of proper convergence criteria. In this paper, three efficient iterative eigenvalue algorithms are considered, i.e., subspace iteration method, iterative Ritz vector method and iterative Lanczos method based on the cell sparse fast solver and loop-unrolling. They are examined under the mode error criterion, i.e., the ratio of the out-of-balance nodal forces and the maximum elastic nodal point forces. Averagely speaking, the iterative Ritz vector method is the most efficient one among the three. Based on the mode error convergence criteria, the eigenvalue solvers are shown to be more stable than those based on eigenvalues only. Compared with ANSYS's subspace iteration and block Lanczos approaches, the subspace iteration presented here appears to be more efficient, while the Lanczos approach has roughly equal efficiency. The methods proposed are robust and efficient. Large size tests show that the improvement in terms of CPU time and storage is tremendous. Also reported is an aggressive shifting technique for the subspace iteration method, based on the mode error convergence criteria. A backward technique is introduced when the shift is not located in the right region. The efficiency of such a technique was demonstrated in the numerical tests.