• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.129-142
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• 1998

In this paper, we propose a blocked variant of the Conjugate Gradient method which performs as well as and has coarser parallelism than the classical Conjugate Gradient method.

• Journal of the military operations research society of Korea
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• v.24 no.1
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• pp.146-156
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• 1998

Cholesky factorization is known to be inefficient to problems with dense column and network problems in interior point methods. We use the conjugate gradient method and preconditioners to improve the convergence rate of the conjugate gradient method. Several preconditioners were applied to LPABO 5.1 and the results were compared with those of CPLEX 3.0. The conjugate gradient method shows to be more efficient than Cholesky factorization to problems with dense columns and network problems. The incomplete Cholesky factorization preconditioner shows to be the most efficient among the preconditioners.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1411-1429
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• 2016

We consider the nite volume element method for elliptic problems using isoparametric bilinear elements on quadrilateral meshes. A gradient recovery method is presented by using the patch interpolation technique. Based on some superclose estimates, we prove that the recovered gradient $R({\nabla}u_h)$ possesses the superconvergence: ${\parallel}{\nabla}u-R({\nabla}u_h){\parallel}=O(h^2){\parallel}u{\parallel}_3$. Finally, some numerical examples are provided to illustrate our theoretical analysis.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.21 no.2
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• pp.189-194
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• 2020

This paper analyzes the gradient descent method, which is the one most used for learning neural networks. Learning means updating a parameter so the loss function is at its minimum. The loss function quantifies the difference between actual and predicted values. The gradient descent method uses the slope of the loss function to update the parameter to minimize error, and is currently used in libraries that provide the best deep learning algorithms. However, these algorithms are provided in the form of a black box, making it difficult to identify the advantages and disadvantages of various gradient descent methods. This paper analyzes the characteristics of the stochastic gradient descent method, the momentum method, the AdaGrad method, and the Adadelta method, which are currently used gradient descent methods. The experimental data used a modified National Institute of Standards and Technology (MNIST) data set that is widely used to verify neural networks. The hidden layer consists of two layers: the first with 500 neurons, and the second with 300. The activation function of the output layer is the softmax function, and the rectified linear unit function is used for the remaining input and hidden layers. The loss function uses cross-entropy error.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1998.06a
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• pp.740-749
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• 1998

This paper proposes compression of image data using neural networks based on conjugate gradient method and dynamic tunneling system. The conjugate gradient method is applied for high speed optimization .The dynamic tunneling algorithms, which is the deterministic method with tunneling phenomenon, is applied for global optimization. Converging to the local minima by using the conjugate gradient method, the new initial point for escaping the local minima is estimated by dynamic tunneling system. The proposed method has been applied the image data compression of 12 ${\times}$12 pixels. The simulation results shows the proposed networks has better learning performance , in comparison with that using the conventional BP as learning algorithm.

• Journal of Electrical Engineering and Technology
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• v.9 no.6
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• pp.1921-1927
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• 2014

A new method for optimizing the magnetic field gradient in the exciting coil of electronic anti-fouling (EAF) system is presented based on changing exciting coil size. In the proposed method, two optimization expressions are deduced based on biot-savart law. The optimization expressions, which can describe the distribution of the magnetic field gradient in the coil, are the function of coil radius and coil length. These optimization expressions can be used to obtain an accurate coil size if the magnetic field gradient on a certain point on the coil's axis of symmetry is needed to be the maximum value. Comparing with the experimental results and the computation results using Finite Element Method simulation to the magnetic field gradient on the coil's axis of symmetry, the computation results obtained by the optimization expression in this article can fit the experimental results and the Finite Element Method results very well. This new method can optimize the EAF system's anti-fouling performance based on improving the magnetic field gradient distribution in the exciting coil.

• Transactions of Materials Processing
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• v.15 no.8 s.89
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• pp.544-549
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• 2006

In this work, we have employed the strain gradient plasticity theory to investigate the effect of material size on the deformation behavior in metal forming process. Flow stress is expressed in terms of strain, strain gradient (spatial derivative of strain) and intrinsic material length. The least square method coupled with strain gradient plasticity was used to calculate the components of strain gradient at each element of material. For demonstrating the size effect, the proposed approach has been applied to plane compression process and micro rolling process. Results show when the characteristic length of the material comes to the intrinsic material length, the effect of strain gradient is noteworthy. For the microcompression, the additional work hardening at higher strain gradient regions results in uniform distribution of strain. In the case of micro-rolling, the strain gradient is remarkable at the exit section where the actual reduction of the rolling finishes and subsequently strong work hardening take places at the section. This results in a considerable increase in rolling force. Rolling force with the strain gradient plasticity considered in analysis increases by 20% compared to that with conventional plasticity theory.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.63-70
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• 2010

In this paper, we develop a nonmonotone spectral memory gradient method for unconstrained optimization, where the spectral stepsize and a class of memory gradient direction are combined efficiently. The global convergence is obtained by using a nonmonotone line search strategy and the numerical tests are also given to show the efficiency of the proposed algorithm.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.20 no.8
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• pp.534-540
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• 2008

In this paper, the optimal bang-bang control problem of an air-conditioning system with slab thermal storage was formulated by gradient method. Furthermore, the numeric solution obtained by gradient method algorithm was compared with the analytic solution obtained on the basis of maximum principle. The control variable is changed uncontinuously at the start time of thermal storage operation in an analytic solution. On the other hand, it is showed as a continuous solution in a numeric solution. The numeric solution reproduces the analytic solution when a tolerance for convergence is applied severely. It is conceivable that gradient method is effective in the analysis of the optimal bang-bang control of the large-scale system like an air-conditioning system with slab thermal storage.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.162-172
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• 2021

This paper proposes stochastic methods to find an approximate solution for the L²-Wasserstein least squares problem of Gaussian measures. The variable for the problem is in a set of positive definite matrices. The first proposed stochastic method is a type of classical stochastic gradient methods combined with projection and the second one is a type of variance reduced methods with projection. Their global convergence are analyzed by using the framework of proximal stochastic gradient methods. The convergence of the classical stochastic gradient method combined with projection is established by using diminishing learning rate rule in which the learning rate decreases as the epoch increases but that of the variance reduced method with projection can be established by using constant learning rate. The numerical results show that the present algorithms with a proper learning rate outperforms a gradient projection method.