• Geophysics and Geophysical Exploration
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• v.25 no.4
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• pp.242-251
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• 2022

Automatic differentiation automatically calculates the derivatives of a function using the chain rule once the forward operation of a function is defined. Given the recent development of computing libraries that support automatic differentiation, many researchers have adopted automatic differentiation techniques to solve geophysical inverse problems. We analyzed the advantages, disadvantages, and performances of automatic differentiation techniques using the gradient calculations of seismic full waveform inversion objective functions. The gradients of objective functions can be expressed as multiplications of the derivatives of the model parameters, wavefields, and objective functions using the chain rule. Using numerical examples, we demonstrated the speed of analytic differentiation and the convenience of complex gradient calculations for automatic differentiation. We calculated derivatives of model parameters and objective functions using automatic differentiation and derivatives of wavefields using analytic differentiation.

• Journal of the Korean Vacuum Society
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• v.10 no.4
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• pp.424-430
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• 2001

An electron Boltzmann equation is solved numerically to calculate the electron energy distribution functions in plasma discharge which is generated by radio-frequency (RF) and microwave frequency electric field. The maintenance field strengths are determined self-consistently by solving the homogeneous electron Boltzmann equation in the Lorentz approximation expressed by 2nd order differential equation and an additional particle balance equation expressed by integro-differential equation. By using this numerical code, the electron energy distribution functions in argon discharge are calculated in the range from RF to microwave frequency. The influence of frequency of the HF electric field on the electron energy distribution functions and ionization rate are investigated.

• Journal of Digital Contents Society
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• v.5 no.4
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• pp.289-294
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• 2004

For the purpose of image analysis, we usually take the application method relying on the various mathematical theories. On the respect of image as two variable function one may uses the gradient vector or several type of energy functions induced by the conventional (partial) derivative. We also have used the tangent plane or curvature vector from the concept of differential geometry {**]. However, these mathematical tools my assume that the given function should be sufficiently smoothing enough to depict every local variation continuously. But the real application of these mathematical methods to the natural images or phenomena may occur the ill-posed problem. In this paper, we have defined the weak derivative as a loose form of the derivative so that it my applied to the irregular case with less ill-posed problem.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.56 no.8
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• pp.1498-1502
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• 2007

There have been many different definitions of energy functions as the second statistics of a signal. In this paper, using the higher order differential energy function, we propose an algorithm separating the amplitude and frequency components in a discrete sinusoidal signal. The proposed amplitude and frequency estimation methods have less computational requirement than the existing methods. It also shows large computational advantage over the root mean square (RMS) calculation of a signal. The proposed methods can be used in the detection of abnormal events in signals on the power line. Computer simulations show that proposed frequency estimation method can detect the presence of voltage increase or decrease for a short period of time. Also, the proposed estimation methods have been compared with existing methods in terms of estimation error variance.

• Journal of Digital Convergence
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• v.14 no.11
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• pp.249-255
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• 2016

Since the elderly voices include a lot of noise caused by physiological changes in respiration, phonation, and resonance, the performance of the convergence health-care equipments such as speech recognition, synthesis, analysis program done by elderly voice is deteriorated. Therefore it is necessary to develop researches to operate health-care instruments with elderly voices. In this study, a voice activity detection using a symmetric higher-order differential energy function (SHODEO) was developed and was compared with auto-correlation function(ACF) and the average magnitude difference function(AMDF). It was confirmed to have a better performance than other methods in the voice interval detection. The voice activity detection will be applied to a voice interface for the elderly to improve the accessibility of the smart devices.

• Proceedings of the KIEE Conference
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• 2005.07c
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• pp.2082-2084
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• 2005

I-V 특성 곡선의 2차 미분을 통해서 얻어지는 전자 에너지 분포 함수를 정확하게 구하기 위해서는 스무딩 과정이 반드시 필요하다. 대표적인 스무딩 방법으로 가우시안 확률 밀도 함수를 instrument함수로 이용하는 가우시안 스무딩이 있다. 본 연구에서는 시스템에 따라서 instrument함수가 다르다는 점에 착안하여, 여러 가지 다른 종류의 확률 밀도 함수를 instrument함수로 사용 스무딩에 적용하여 확률 밀도 함수에 따른 노이즈 제거 및 전자 에너지 분포 함수의 정확도를 비교하였고. 동시에 대표적인 범용 스무딩 방법인 사비츠키-골래이 스무딩, Polynomial fitting과도 그 결과를 비교 분석하였다.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.21 no.4
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• pp.391-399
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• 2008

In this paper, we further generalize the work of Lin and Abel to the case of the first and the second order derivatives of energy release rates for two-dimensional, multiply cracked systems. The direct integral expressions are presented for the energy release rates and their first and second order derivatives. The salient feature of this numerical method is that the energy release rates and their first and second order derivatives can be computed in a single analysis. It is demonstrated through a set of examples that the proposed method gives expectedly decreasing, but acceptably accurate results for the energy release rates and their first and second order derivatives. The computed errors were approximately 0.5% for the energy release rates, $3\sim5%$ for their first order derivatives and $10\sim20%$ for their second order derivatives for the mesh densities used in the examples. Potential applications of the present method include a universal size effect model and a probabilistic fracture analysis of cracked structures.

• Geophysics and Geophysical Exploration
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• v.26 no.3
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• pp.114-125
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• 2023

The physics-informed neural network (PINN) has been proposed to overcome the limitations of various numerical methods used to solve partial differential equations (PDEs) and the drawbacks of purely data-driven machine learning. The PINN directly applies PDEs to the construction of the loss function, introducing physical constraints to machine learning training. This technique can also be applied to wave equation modeling. However, to solve the wave equation using the PINN, second-order differentiations with respect to input data must be performed during neural network training, and the resulting wavefields contain complex dynamical phenomena, requiring careful strategies. This tutorial elucidates the fundamental concepts of the PINN and discusses considerations for wave equation modeling using the PINN approach. These considerations include spatial coordinate normalization, the selection of activation functions, and strategies for incorporating physics loss. Our experimental results demonstrated that normalizing the spatial coordinates of the training data leads to a more accurate reflection of initial conditions in neural network training for wave equation modeling. Furthermore, the characteristics of various functions were compared to select an appropriate activation function for wavefield prediction using neural networks. These comparisons focused on their differentiation with respect to input data and their convergence properties. Finally, the results of two scenarios for incorporating physics loss into the loss function during neural network training were compared. Through numerical experiments, a curriculum-based learning strategy, applying physics loss after the initial training steps, was more effective than utilizing physics loss from the early training steps. In addition, the effectiveness of the PINN technique was confirmed by comparing these results with those of training without any use of physics loss.

• Journal of Energy Engineering
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• v.4 no.3
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• pp.331-337
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• 1995

석탄가스화 복합발전 시스템은 고효율 및 우수한 환경보전성으로 인하여 차세대 화력발전 시스템으로 각광받고 있다. 본 연구는 석탄가스화기내의 비반응 유동장에 대한 연구로서, 순수유동장 및 미분탄입자를 포함한 이상유동장에 대한 전산해석을 수행하였다. 가스화기내의 유동장을 기술하는 Navier-Stokes방정식을 유한차분법에 의하여 해석하고 그 결과를 나타내었다. 특히 선회유동의 영향에 의한 미분탄입자의 거동 및 재순환영역의 특성에 대하여 선회강도의 함수로 고찰하였다. 해석결과에 의하면 가스화기내에서는 며책의 재순환영역이 형성됨을 알 수 있었다. 비반응유동장의 해석결과이지만, 선회유동은 화염안정화에 긍정적인 영향을 줄 수 있을 것으로 추측되는 결과를 보였다.

• Journal of the KSME
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• v.23 no.3
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• pp.200-206
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• 1983

FAM의 기본적인 구상은 해석 하고자하는 선형 또는 비선형 편미분 방정식을 국부적으로 해석 적인 해를 구하여 이용하자는 것이다. 그러기 위하여 유한차분법(FDM)과 유한변분법(FEM)에 서와 같이 전체유동장을 작은 요소로 나누고 그 요소 내에서 국부해를 구한 다음 이들 요소를 중첩시킴으로써 각 요소의 미지수에 대한 대수식을 얻어서 수치해를 구하자는 것이다. 그러나 FDM에서와 같이 국부요소에서 미분항을 구하지 않고, FEM 에서와 같이 요소에서 형상함수를 도입하지 않는 상태에서 해석적인 해를 구하고 있기 때문에 수치해석에서 얻어지는 미분양들은 비교적 정확하게 구해진다. 따라서 Navier-Stokes 방정식이나 에너지 방정식에서 최고차항이 작은 파라메타, 즉 레이놀즈수나 피크리수의 역수로 곱하여서 있는 경우에도 안정된 해를 구할 수 있다고 알려져 있다. 요소자체의 계수를 구하는 데는 계산시간이 많이 소요되지만 수치해석 상의 안정성이나 수렴성이 좋기 때문에 전체계산시간은 오히려 적게 걸리는 경우도 있다고 한다.