• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.215-228
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• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• ETRI Journal
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• v.41 no.3
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• pp.298-307
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• 2019

This paper investigates the use of the inverse-free sparse Bayesian learning (SBL) approach for peak-to-average power ratio (PAPR) reduction in orthogonal frequency-division multiplexing (OFDM)-based multiuser massive multiple-input multiple-output (MIMO) systems. The Bayesian inference method employs a truncated Gaussian mixture prior for the sought-after low-PAPR signal. To learn the prior signal, associated hyperparameters and underlying statistical parameters, we use the variational expectation-maximization (EM) iterative algorithm. The matrix inversion involved in the expectation step (E-step) is averted by invoking a relaxed evidence lower bound (relaxed-ELBO). The resulting inverse-free SBL algorithm has a much lower complexity than the standard SBL algorithm. Numerical experiments confirm the substantial improvement over existing methods in terms of PAPR reduction for different MIMO configurations.

• Nuclear Engineering and Technology
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• v.54 no.4
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• pp.1230-1244
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• 2022

This paper proposes an algorithm for signal validation using unsupervised methods in emergency situations at nuclear power plants (NPPs) when signals are rapidly changing. The algorithm aims to determine the stuck failures of signals in real time based on a variational auto-encoder (VAE), which employs unsupervised learning, and long short-term memory (LSTM). The application of unsupervised learning enables the algorithm to detect a wide range of stuck failures, even those that are not trained. First, this paper discusses the potential failure modes of signals in NPPs and reviews previous studies conducted on signal validation. Then, an algorithm for detecting signal failures is proposed by applying LSTM and VAE. To overcome the typical problems of unsupervised learning processes, such as trainability and performance issues, several optimizations are carried out to select the inputs, determine the hyper-parameters of the network, and establish the thresholds to identify signal failures. Finally, the proposed algorithm is validated and demonstrated using a compact nuclear simulator.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1441-1462
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• 2018

In this paper, we investigate the fractional p&q-Kirchhoff type system $$\{M_1([u]^p_{s,p})(-{\Delta})^s_pu+V_1(x){\mid}u{\mid}^{p-2}u\\{\hfill{10}}={\ell}k^{-1}F_u(x,\;u,\;v)+{\lambda}{\alpha}(x){\mid}u{\mid}^{m-2}u,\;x{\in}{\Omega}\\M_2([u]^q_{s,q})(-{\Delta})^s_qv+V_2(x){\mid}v{\mid}^{q-2}v\\{\hfill{10}}={\ell}k^{-1}F_v(x,u,v)+{\mu}{\alpha}(x){\mid}v{\mid}^{m-2}v,\;x{\in}{\Omega},\\u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}{\subset}{\mathbb{R}}^N$ is an unbounded domain with smooth boundary ${\partial}{\Omega}$, and $0<s<1<p{\leq}q$ and sq < N, ${\lambda},{\mu}>0$, $1<m{\leq}k<p^*_s$, ${\ell}{\in}R$, while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (1.2) below. $V_1(x)$, $V_2(x)$, $a(x):{\mathbb{R}}^N{\rightarrow}(0,\;{\infty})$ are three positive weights, $M_1$, $M_2$ are continuous and positive functions in ${\mathbb{R}}^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.

• Nuclear Engineering and Technology
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• v.55 no.5
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• pp.1630-1643
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• 2023

The correct situation awareness (SA) of operators is important for managing nuclear power plants (NPPs), particularly in accident-related situations. Among the three levels of SA suggested by Ensley, Level 3 SA (i.e., projection of the future status of the situation) is challenging because of the complexity of NPPs as well as the uncertainty of accidents. Hence, several prediction methods using artificial intelligence techniques have been proposed to assist operators in accident prediction. However, these methods only predict short-term plant status (e.g., the status after a few minutes) and do not provide information regarding the uncertainty associated with the prediction. This paper proposes an algorithm that can predict the multivariate and long-term behavior of plant parameters for 2 h with 120 steps and provide the uncertainty of the prediction. The algorithm applies bidirectional long short-term memory and an attention mechanism, which enable the algorithm to predict the precise long-term trends of the parameters with high prediction accuracy. A conditional variational autoencoder was used to provide uncertainty information about the network prediction. The algorithm was trained, optimized, and validated using a compact nuclear simulator for a Westinghouse 900 MWe NPP.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.663-677
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• 1999

This work is concerned mainly with the variational problem on an immersion x:M $\rightarrow$} $E^4$ . A new approach is introduced depends on the normal variation in any arbitrary normal direction in the normal bundle. The results of this work are considered as a continuation and an extension to that obtained in [1], [2] and [3], [4] respectively. The methods adapted here are based on Cartan's methods of moving frames and the calculus of variations.

• The Pure and Applied Mathematics
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• v.16 no.1
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• pp.85-98
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• 2009

Strong convergence theorems on viscosity approximation methods for finite nonexpansive mappings are established in Banach spaces. The main theorem generalize the corresponding result of Kim and Xu [10] to the viscosity approximation method for finite nonexpansive mappings in a reflexive Banach space having a uniformly Gateaux differentiable norm. Our results also improve the corresponding results of [7, 8, 19, 20].

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.1 no.1
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• pp.1-34
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• 1997

In this study, various available meshless methods are briefly reviewed and the connection among them is investigated. The objective of meshless methods is to eliminate some difficulties which are originated from reliance on a mesh by constructing the approximation entirely in terms of nodes. Especially, focusing on Element Free Galerkin Method(EFGM) based on moving least square interpolants(MLSI), a new implementation is developed based on a variational principle with penalty function method were used to enforce the essential boundary condition. In addition, the weighted orthogonal basis functions are constructed to overcome disadvantage of MLSI.

• Journal of Korean Society for Quality Management
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• v.49 no.4
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• pp.581-594
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• 2021

Purpose: The purpose of this study is to compare machine learning models for anomaly detection of mechanical facility equipment and suggest an anomaly detection system for mechanical facility equipment in subway stations. It helps to predict failures and plan the maintenance of facility. Ultimately it aims to improve the quality of facility equipment. Methods: The data collected from Daejeon Metropolitan Rapid Transit Corporation was used in this experiment. The experiment was performed using Python, Scikit-learn, tensorflow 2.0 for preprocessing and machine learning. Also it was conducted in two failure states of the equipment. We compared and analyzed five unsupervised machine learning models focused on model Long Short-Term Memory Variational Autoencoder(LSTM-VAE). Results: In both experiments, change in vibration and current data was observed when there is a defect. When the rotating body failure was happened, the magnitude of vibration has increased but current has decreased. In situation of axis alignment failure, both of vibration and current have increased. In addition, model LSTM-VAE showed superior accuracy than the other four base-line models. Conclusion: According to the results, model LSTM-VAE showed outstanding performance with more than 97% of accuracy in the experiments. Thus, the quality of mechanical facility equipment will be improved if the proposed anomaly detection system is established with this model used.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1151-1164
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• 2009

Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\rightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $\rightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${\neq}{\emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, ($n{\geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\lambda_n$} $\subset$ (0, 1) satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n$ = 0 and $\sum_{n=0}^{\infty}{\lambda}_n={\infty}$, {$\beta_n$} $\subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.