Search results for: Optimization method

Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

Abstract:

Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, Step method, delay differential equation, simulation.

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Authors: Kriuk Boris, Kriuk Fedor

Abstract:

This paper presents a comparative study of fundamental similarity functions for Siamese networks in semantic textual similarity (STS) tasks. We evaluate various similarity functions using the STS Benchmark dataset, analyzing their performance and stability. Additionally, we present a multi-objective approach for optimal threshold selection. Our findings provide insights into the effectiveness of different similarity functions and offer a straightforward method for threshold selection optimization, contributing to the advancement of Siamese network architectures in STS applications.

Keywords: Siamese networks, Semantic textual similarity, Similarity functions, STS Benchmark dataset, Threshold selection.

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Authors: Hidehiko Okada

Abstract:

The author previously proposed an extension of differential evolution. The proposed method extends the processes of DE to handle interval numbers as genotype values so that DE can be applied to interval-valued optimization problems. The interval DE can employ either of two interval models, the lower and upper model or the center and width model, for specifying genotype values. Ability of the interval DE in searching for solutions may depend on the model. In this paper, the author compares the two models to investigate which model contributes better for the interval DE to find better solutions. Application of the interval DE is evolutionary training of interval-valued neural networks. A result of preliminary study indicates that the CW model is better than the LU model: the interval DE with the CW model could evolve better neural networks. 

Keywords: Evolutionary algorithms, differential evolution, neural network, neuroevolution, interval arithmetic.

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Authors: R. Rajabpour, N. Talebbeydokhti, M. H. Ahmadi

Abstract:

In recent years, new techniques for solving complex problems in engineering are proposed. One of these techniques is JPSO algorithm. With innovative changes in the nature of the jump algorithm JPSO, it is possible to construct a graph-based solution with a new algorithm called G-JPSO. In this paper, a new algorithm to solve the optimal control problem Fletcher-Powell and optimal control of pumps in water distribution network was evaluated. Optimal control of pumps comprise of optimum timetable operation (status on and off) for each of the pumps at the desired time interval. Maximum number of status on and off for each pumps imposed to the objective function as another constraint. To determine the optimal operation of pumps, a model-based optimization-simulation algorithm was developed based on G-JPSO and JPSO algorithms. The proposed algorithm results were compared well with the ant colony algorithm, genetic and JPSO results. This shows the robustness of proposed algorithm in finding near optimum solutions with reasonable computational cost.

Keywords: G-JPSO, operation, optimization, pumping station, water distribution networks.

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Authors: Jafar Biazar, Behzad Ghanbari

Abstract:

In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.

Keywords: System of nonlinear equations.

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Authors: A. Soltani, M. Karimi Demneh

Abstract:

In this paper, modeling of an acoustic enclosed vehicle cabin has been carried out by using boundary element method. Also, the second purpose of this study is analyzing of linear wave equation in an acoustic field. The resultants of this modeling consist of natural frequencies that have been compared with resultants derived from finite element method. By using numerical method (boundary element method) and after solution of wave equation inside an acoustic enclosed cabin, this method has been progressed to simulate noise inside a simple vehicle cabin.

Keywords: Boundary element method, natural frequency, noise, vehicle cabin.

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Authors: B. S. Raghavendra, D. Narayana Dutt

Abstract:

In this paper, we have developed a method to compute fractal dimension (FD) of discrete time signals, in the time domain, by modifying the box-counting method. The size of the box is dependent on the sampling frequency of the signal. The number of boxes required to completely cover the signal are obtained at multiple time resolutions. The time resolutions are made coarse by decimating the signal. The loglog plot of total number of boxes required to cover the curve versus size of the box used appears to be a straight line, whose slope is taken as an estimate of FD of the signal. The results are provided to demonstrate the performance of the proposed method using parametric fractal signals. The estimation accuracy of the method is compared with that of Katz, Sevcik, and Higuchi methods. In addition, some properties of the FD are discussed.

Keywords: Box-counting, Fractal dimension, Higuchi method, Katz method, Parametric fractal signals, Sevcik method.

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Authors: Yiqin Lin, Liang Bao, Qinghua Wu, Liping Zhou

Abstract:

In this paper, we propose a direct method based on the real Schur factorization for solving the projected Sylvester equation with relatively small size. The algebraic formula of the solution of the projected continuous-time Sylvester equation is presented. The computational cost of the direct method is estimated. Numerical experiments show that this direct method has high accuracy.

Keywords: Projected Sylvester equation, Schur factorization, Spectral projection, Direct method.

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Authors: Vinod Kumar Khera, R. K. Sharma, A. K. Gupta

Abstract:

Power dissipation increases exponentially during test mode as compared to normal operation of the circuit. In extreme cases, test power is more than twice the power consumed during normal operation mode. Test vector generation scheme is key component in deciding the power hungriness of a circuit during testing. Test vector count and consequent leakage current are functions of test vector generation scheme. Fault based test vector count optimization has been presented in this work. It helps in reducing test vector count and the leakage current. In the presented scheme, test vectors have been reduced by extracting essential child vectors. The scheme has been tested experimentally using stuck at fault models and results ensure the reduction in test vector count.

Keywords: Low power VLSI testing, independent fault, essential faults, test vector reduction.

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Authors: Miroslav Byrtus

Abstract:

Large rotating systems, especially gear drives and gearboxes, occur as parts of many mechanical devices transmitting the torque with relatively small loss of power. With the increased demand for high speed machinery, mathematical modeling and dynamic analysis of gear drives gained importance. Mathematical description of such mechanical systems is a complex task evolving for several decades. In gear drive dynamic models, which include flexible shafts, bearings and gearing and use the finite elements, nonlinear effects due to gear mesh and bearings are usually ignored, for such models have large number of degrees of freedom (DOF) and it is computationally expensive to analyze nonlinear systems with large number of DOF. Therefore, these models are not suitable for simulation of nonlinear behavior with amplitude jumps in frequency response. The contribution uses a methodology of nonlinear large rotating system modeling which is based on degrees of freedom (DOF) number reduction using modal synthesis method (MSM). The MSM enables significant DOF number reduction while keeping the nonlinear behavior of the system in a specific frequency range. Further, the MSM with DOF number reduction is suitable for including detail models of nonlinear couplings (mainly gear and bearing couplings) into the complete gear drive models. Since each subsystem is modeled separately using different FEM systems, it is advantageous to parameterize models of subsystems and to use the parameterization for optimization of chosen design parameters. Final complex model of gear drive is assembled in MATLAB and MATLAB tools are used for dynamical analysis of the nonlinear system. The contribution is further focused on developing of a methodology for investigation of behavior of the system by Nonlinear Normal Modes with combination of the MSM using numerical continuation method. The proposed methodology will be tested using a two-stage gearbox including its housing.

Keywords: Vibro-impact system, rotating system, gear drive, modal synthesis method, numerical continuation method, periodic solution.

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Authors: S. J. Hosseini Ghoncheh, M. Paripour

Abstract:

In this paper the Huang-s method for solving a m×n fuzzy linear system when, m≤ n, is considered. The method in detail is discussed and illustrated by solving some numerical examples.

Keywords: Fuzzy number, fuzzy linear systems, Huang's method.

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Authors: M. Zarebnia, R. Parvaz

Abstract:

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

Keywords: Benjamin-Bona-Mahony-Burgers equation, Cubic Bspline, Collocation method, Finite difference.

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Authors: Muammer Ozkan, Orkun Ozener, Irfan Yavasliol

Abstract:

The fuel cost of the motor vehicle operating on its common route is an important part of the operating cost. Therefore, the importance of the fuel saving is increasing day by day. One of the parameters which improve fuel saving is the regulation of driving characteristics. The number and duration of stop is increased by the heavy traffic load. It is possible to improve the fuel saving with regulation of traffic flow and driving characteristics. The researches show that the regulation of the traffic flow decreases fuel consumption, but it is not enough to improve fuel saving without the regulation of driving characteristics. This study analyses the fuel consumption of two trips of city bus operating on its common route and determines the effect of traffic density and driving characteristics on fuel consumption. Finally it offers some suggestions about regulation of driving characteristics to improve the fuel saving. Fuel saving is determined according to the results obtained from simulation program. When experimental and simulation results are compared, it has been found that the fuel saving was reached up the to 40 percent ratios.

Keywords: Fuel Consumption, Fuel Economy, Driving Characteristics, Optimization

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Authors: Dong-Chul Lee, Byung-Kwan Oh, Se-Woon Choi, Hyo-Sun Park

Abstract:

The unanticipated brittle fracture of connection of the steel moment resisting frame (SMRF) occurred in 1994 the Northridge earthquake. Since then, the researches for the vulnerability of connection of the existing SMRF and for rehabilitation of those buildings were conducted. This paper suggests performance-based optimal seismic retrofit technique using connection upgrade. For optimal design, a multi-objective genetic algorithm(NSGA-II) is used. One of the two objective functions is to minimize initial cost and another objective function is to minimize lifetime seismic damages cost. The optimal algorithm proposed in this paper is performed satisfying specified performance objective based on FEMA 356. The nonlinear static analysis is performed for structural seismic performance evaluation. A numerical example of SAC benchmark SMRF is provided using the performance-based optimal seismic retrofit technique proposed in this paper

Keywords: connection upgrade, performace-based seismicdesign, seismic retrofit, multi-objective optimization

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Authors: Jain-Shing Liu

Abstract:

In this paper, we present a matrix game-theoretic cross-layer optimization formulation to maximize the network lifetime in wireless ad hoc networks with network coding. To this end, we introduce a cross-layer formulation of general NUM (network utility maximization) that accommodates routing, scheduling, and stream control from different layers in the coded networks. Specifically, for the scheduling problem and then the objective function involved, we develop a matrix game with the strategy sets of the players corresponding to hyperlinks and transmission modes, and design the payoffs specific to the lifetime. In particular, with the inherit merit that matrix game can be solved with linear programming, our cross-layer programming formulation can benefit from both game-based and NUM-based approaches at the same time by cooperating the programming model for the matrix game with that for the other layers in a consistent framework. Finally, our numerical example demonstrates its performance results on a well-known wireless butterfly network to verify the cross-layer optimization scheme.

Keywords: Cross-layer design, Lifetime maximization, Matrix game, Network coding

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Authors: Abdulqader M. Mohsen, Ahamad Tajudin Khader, Dhanesh Ramachandram, Abdullatif Ghallab

Abstract:

The physical methods for RNA secondary structure prediction are time consuming and expensive, thus methods for computational prediction will be a proper alternative. Various algorithms have been used for RNA structure prediction including dynamic programming and metaheuristic algorithms. Musician's behaviorinspired harmony search is a recently developed metaheuristic algorithm which has been successful in a wide variety of complex optimization problems. This paper proposes a harmony search algorithm (HSRNAFold) to find RNA secondary structure with minimum free energy and similar to the native structure. HSRNAFold is compared with dynamic programming benchmark mfold and metaheuristic algorithms (RnaPredict, SetPSO and HelixPSO). The results showed that HSRNAFold is comparable to mfold and better than metaheuristics in finding the minimum free energies and the number of correct base pairs.

Keywords: Metaheuristic algorithms, dynamic programming algorithms, harmony search optimization, RNA folding, Minimum free energy.

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Authors: Kolar Jan, Dvorak Vaclav

Abstract:

In this study, an optimization of supersonic air-to-air ejector is carried out by a recently developed single-objective genetic algorithm based on adaption of sequence of individuals. Adaptation of sequence is based on Shape-based distance of individuals and embedded micro-genetic algorithm. The optimal sequence found defines the succession of CFD-aimed objective calculation within each generation of regular micro-genetic algorithm. A spring-based deformation mutates the computational grid starting the initial individualvia adapted population in the optimized sequence. Selection of a generation initial individual is knowledge-based. A direct comparison of the newly defined and standard micro-genetic algorithm is carried out for supersonic air-to-air ejector. The only objective is to minimize the loose of total stagnation pressure in the ejector. The result is that sequence-adopted micro-genetic algorithm can provide comparative results to standard algorithm but in significantly lower number of overall CFD iteration steps.

Keywords: Grid deformation, Micro-genetic algorithm, shapebased sequence, supersonic ejector.

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Authors: K. Djellabi, M. E. H. Latreche

Abstract:

Induction heating computer simulation is a powerful tool for process design and optimization, induction coil design, equipment selection, as well as education and business presentations. The authors share their vast experience in the practical use of computer simulation for different induction heating and heat treating processes. In this paper treated with mathematical modeling and numerical simulation of induction heating furnaces with axisymmetric geometries for the numerical solution, we propose finite element methods combined with boundary (FEM) for the electromagnetic model using COMSOL® Multiphysics Software. Some numerical results for an industrial furnace are shown with high frequency.

Keywords: Numerical methods, Induction furnaces, Induction Heating, Finite element method, Comsol Multiphysics software.

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Authors: A. D. Parekh, P. R. Tailor, H.R Jivanramajiwala

Abstract:

The present work deals with optimization of cascade refrigeration system using eco friendly refrigerants pair R507A and R23. R507A is azeotropic mixture composed of HFC refrigerants R125/R143a (50%/50% by wt.). R23 is a single component HFC refrigerant used as replacement to CFC refrigerant R13 in low temperature applications. These refrigerants have zero ozone depletion potential and are non-flammable. Optimization of R507AR23 cascade refrigeration system performance parameters such as minimum work required, refrigeration effect, coefficient of performance and exergetic efficiency was carried out in terms of eight operating parameters- combinations using Genetic Algorithm tool. The eight operating parameters include (1) low side evaporator temperature (2) high side condenser temperature (3) temperature difference in the cascade heat exchanger (4) low side condenser temperature (5) low side degree of subcooling (6) high side degree of subcooling (7) low side degree of superheating (8) high side degree of superheating. Results show that for minimum work system should operate at high temperature in low side evaporator, low temperature in high side condenser, low temperature difference in cascade condenser, high temperature in low side condenser and low degree of subcooling and superheating in both side. For maximum refrigeration effect system should operate at high temperature in low side evaporator, high temperature in high side condenser, high temperature difference in cascade condenser, low temperature in low side condenser and higher degree of subcooling in LT and HT side. For maximum coefficient of performance and exergetic efficiency, system should operate at high temperature in low side evaporator, low temperature in high side condenser, low temperature difference in cascade condenser, high temperature in low side condenser and higher degree of subcooling and superheating in low side of the system.

Keywords: Cascade refrigeration system, Genetic Algorithm, R507A, R23,

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Authors: Abbas Al-Refaie

Abstract:

This paper aims at improving the performance of the tableting process using statistical quality control and fuzzy goal programming. The tableting process was studied. Statistical control tools were used to characterize the existing process for three critical responses including the averages of a tablet’s weight, hardness, and thickness. At initial process factor settings, the estimated process capability index values for the tablet’s averages of weight, hardness, and thickness were 0.58, 3.36, and 0.88, respectively. The L₉ array was utilized to provide experimentation design. Fuzzy goal programming was then employed to find the combination of optimal factor settings. Optimization results showed that the process capability index values for a tablet’s averages of weight, hardness, and thickness were improved to 1.03, 4.42, and 1.42, respectively. Such improvements resulted in significant savings in quality and production costs.

Keywords: Fuzzy goal programming, control charts, process capability, tablet optimization.

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Authors: H. Taghvafard, G. H. Erjaee

Abstract:

A method for solving linear and non-linear Goursat problem is given by using the two-dimensional differential transform method. The approximate solution of this problem is calculated in the form of a series with easily computable terms and also the exact solutions can be achieved by the known forms of the series solutions. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Several examples are given to demonstrate the reliability and the performance of the presented method.

Keywords: Quadrature, Spline interpolation, Trapezoidal rule, Numericalintegration, Error analysis.

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Authors: T. Ganesan, M. S. Aris, I. Elamvazuthi, Momen Kamal Tageldeen

Abstract:

Terms set in power purchase agreements (PPA) challenge power utility companies in balancing between the returns (from maximizing power production) and securing long term supply contracts at capped production. The production limitation set in the PPA has driven efforts to maximize profits through efficient and economic power production. In this paper, a combined industrial-scale gas turbine (GT) - absorption chiller (AC) system is considered to cool the GT air intake for reducing the plant’s heat rate (HR). This GT-AC system is optimized while considering power output limitations imposed by the PPA. In addition, the proposed formulation accounts for uncertainties in the ambient temperature using Type-2 fuzzy programming. Using the enhanced chaotic differential evolution (CEDE), the Pareto frontier was constructed and the optimization results are analyzed in detail.

Keywords: Absorption chillers, turbine inlet air cooling, power purchase agreement, multiobjective optimization, type-2 fuzzy programming, chaotic differential evolution.

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Authors: Ahmed A. D. Sarhan, Hong Wan Ping, M. Sayuti

Abstract:

In this current era of competitive machinery productions, the industries are designed to place more emphasis on the product quality and reduction of cost whilst abiding by the pollution-preventing policy. In attempting to delve into the concerns, the industries are aware that the effectiveness of existing lubrication systems must be improved to achieve power-efficient and pollution-preventing machining processes. As such, this research is targeted to study on a plausible solution to the issue in grinding titanium alloy (Ti-6Al-4V) by using nanolubrication, as an alternative to flood grinding. The aim of this research is to evaluate the optimum condition of grinding force and surface roughness using MQL lubricating system to deliver nano-oil at different level of weight concentration of Silicon Dioxide (SiO₂) mixed normal mineral oil. Taguchi Design of Experiment (DoE) method is carried out using a standard Taguchi orthogonal array of L16(43) to find the optimized combination of weight concentration mixture of SiO₂, nozzle orientation and pressure of MQL. Surface roughness and grinding force are also analyzed using signal-to-noise(S/N) ratio to determine the best level of each factor that are tested. Consequently, the best combination of parameters is tested for a period of time and the results are compared with conventional grinding method of dry and flood condition. The results show a positive performance of MQL nanolubrication.  

Keywords: Grinding, MQL, precision grinding, Taguchi optimization, titanium alloy.

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Authors: Jatinder Kumar, Vinod Kumar

Abstract:

Ultrasonic machining (USM) is a non-traditional machining process being widely used for commercial machining of brittle and fragile materials such as glass, ceramics and semiconductor materials. However, USM could be a viable alternative for machining a tough material such as titanium; and this aspect needs to be explored through experimental research. This investigation is focused on exploring the use of ultrasonic machining for commercial machining of pure titanium (ASTM Grade-I) and evaluation of tool wear rate (TWR) under controlled experimental conditions. The optimal settings of parameters are determined through experiments planned, conducted and analyzed using Taguchi method. In all, the paper focuses on parametric optimization of ultrasonic machining of pure titanium metal with TWR as response, and validation of the optimized value of TWR by conducting confirmatory experiments.

Keywords: Ultrasonic machining, titanium, tool wear rate

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Authors: Hussam Elias, Ninovic Perez, Holger Hirsch

Abstract:

With ever-increasing use of equipment, device or more generally any electrical or electronic system, the chance of Electromagnetic incompatibility incidents has considerably increased which demands more attention to ensure the possible risks of these technologies. Therefore, complying with certain Electromagnetic compatibility (EMC) rules and not overtaking an acceptable level of radiated emissions are utmost importance for the diffusion of electronic products. In this paper, developed measure tool and a convolutional neural network were used to propose a method to reduce the required time to carry out the final measurement phase of Electromagnetic interference (EMI) measurement according to the norm EN 55032 by predicting the radiated emission and determining the height of the antenna that meets the maximum radiation value.

Keywords: Antenna height, Convolutional Neural Network, Electromagnetic Compatibility, Mean Absolute Error, position error.

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Authors: Rakesh Prajapati, Purvik Patel, Hardik Patel

Abstract:

Productivity and quality are two important aspects that have become great concerns in today’s competitive global market. Every production/manufacturing unit mainly focuses on these areas in relation to the process, as well as the product developed. The electrical discharge machining (EDM) process, even now it is an experience process, wherein the selected parameters are still often far from the maximum, and at the same time selecting optimization parameters is costly and time consuming. Material Removal Rate (MRR) during the process has been considered as a productivity estimate with the aim to maximize it, with an intention of minimizing surface roughness taken as most important output parameter. These two opposites in nature requirements have been simultaneously satisfied by selecting an optimal process environment (optimal parameter setting). Objective function is obtained by Regression Analysis and Analysis of Variance. Then objective function is optimized using Genetic Algorithm technique. The model is shown to be effective; MRR and Surface Roughness improved using optimized machining parameters.

Keywords: Material removal rate, TWR, OC, DOE, ANOVA, MINITAB.

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Authors: A. Elyousfi, A. Tamtaoui, E. Bouyakhf

Abstract:

The H.264/AVC video coding standard contains a number of advanced features. Ones of the new features introduced in this standard is the multiple intramode prediction. Its function exploits directional spatial correlation with adjacent block for intra prediction. With this new features, intra coding of H.264/AVC offers a considerably higher improvement in coding efficiency compared to other compression standard, but computational complexity is increased significantly when brut force rate distortion optimization (RDO) algorithm is used. In this paper, we propose a new fast intra prediction mode decision method for the complexity reduction of H.264 video coding. for luma intra prediction, the proposed method consists of two step: in the first step, we make the RDO for four mode of intra 4x4 block, based the distribution of RDO cost of those modes and the idea that the fort correlation with adjacent mode, we select the best mode of intra 4x4 block. In the second step, we based the fact that the dominating direction of a smaller block is similar to that of bigger block, the candidate modes of 8x8 blocks and 16x16 macroblocks are determined. So, in case of chroma intra prediction, the variance of the chroma pixel values is much smaller than that of luma ones, since our proposed uses only the mode DC. Experimental results show that the new fast intra mode decision algorithm increases the speed of intra coding significantly with negligible loss of PSNR.

Keywords: Intra prediction, H264/AVC, video coding, encodercomplexity.

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Authors: Zhenyu Zhao, Lei You

Abstract:

The numerical analytic continuation of a function f(z) = f(x + iy) on a strip is discussed in this paper. The data are only given approximately on the real axis. The periodicity of given data is assumed. A truncated Fourier spectral method has been introduced to deal with the ill-posedness of the problem. The theoretic results show that the discrepancy principle can work well for this problem. Some numerical results are also given to show the efficiency of the method.

Keywords: Analytic continuation, ill-posed problem, regularization method Fourier spectral method, the discrepancy principle.

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Authors: K. Prajindra Sankar, S.K. Tiong, S.P. Johnny Koh

Abstract:

This paper presents a tested research concept that implements a complex evolutionary algorithm, genetic algorithm (GA), in a multi-microcontroller environment. Parallel Distributed Genetic Algorithm (PDGA) is employed in adaptive beam forming technique to reduce power usage of adaptive antenna at WCDMA base station. Adaptive antenna has dynamic beam that requires more advanced beam forming algorithm such as genetic algorithm which requires heavy computation and memory space. Microcontrollers are low resource platforms that are normally not associated with GAs, which are typically resource intensive. The aim of this project was to design a cooperative multiprocessor system by expanding the role of small scale PIC microcontrollers to optimize WCDMA base station transmitter power. Implementation results have shown that PDGA multi-microcontroller system returned optimal transmitted power compared to conventional GA.

Keywords: Microcontroller, Genetic Algorithm, Adaptiveantenna, Power optimization.

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Authors: Khosrow Maleknejad, Asyieh Ebrahimzadeh

Abstract:

In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential (VID) equation is considered. The method is developed by means of the Legendre wavelet approximation and collocation method. The properties of Legendre wavelet together with Gaussian integration method are utilized to reduce the problem to the solution of nonlinear programming one. Some numerical examples are given to confirm the accuracy and ease of implementation of the method.

Keywords: Collocation method, Legendre wavelet, optimal control, Volterra integro-differential equation.

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