Search results for: Preconditioned Conjugate Gradien

Authors: Z. Zainuddin, W. D. Wan Rosli, R. Lanouette, S. Sathasivam

Abstract:

This paper reports work done to improve the modeling of complex processes when only small experimental data sets are available. Neural networks are used to capture the nonlinear underlying phenomena contained in the data set and to partly eliminate the burden of having to specify completely the structure of the model. Two different types of neural networks were used for the application of pulping problem. A three layer feed forward neural networks, using the Preconditioned Conjugate Gradient (PCG) methods were used in this investigation. Preconditioning is a method to improve convergence by lowering the condition number and increasing the eigenvalues clustering. The idea is to solve the modified odified problem M-1 Ax= M-1b where M is a positive-definite preconditioner that is closely related to A. We mainly focused on Preconditioned Conjugate Gradient- based training methods which originated from optimization theory, namely Preconditioned Conjugate Gradient with Fletcher-Reeves Update (PCGF), Preconditioned Conjugate Gradient with Polak-Ribiere Update (PCGP) and Preconditioned Conjugate Gradient with Powell-Beale Restarts (PCGB). The behavior of the PCG methods in the simulations proved to be robust against phenomenon such as oscillations due to large step size.

Keywords: Convergence, pulping modeling, neural networks, preconditioned conjugate gradient.

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Authors: W. D. Wan Rosli, Z. Zainuddin, R. Lanouette, S. Sathasivam

Abstract:

This paper reports work done to improve the modeling of complex processes when only small experimental data sets are available. Neural networks are used to capture the nonlinear underlying phenomena contained in the data set and to partly eliminate the burden of having to specify completely the structure of the model. Two different types of neural networks were used for the application of Pulping of Sugar Maple problem. A three layer feed forward neural networks, using the Preconditioned Conjugate Gradient (PCG) methods were used in this investigation. Preconditioning is a method to improve convergence by lowering the condition number and increasing the eigenvalues clustering. The idea is to solve the modified problem where M is a positive-definite preconditioner that is closely related to A. We mainly focused on Preconditioned Conjugate Gradient- based training methods which originated from optimization theory, namely Preconditioned Conjugate Gradient with Fletcher-Reeves Update (PCGF), Preconditioned Conjugate Gradient with Polak-Ribiere Update (PCGP) and Preconditioned Conjugate Gradient with Powell-Beale Restarts (PCGB). The behavior of the PCG methods in the simulations proved to be robust against phenomenon such as oscillations due to large step size.

Keywords: Convergence, Modeling, Neural Networks, Preconditioned Conjugate Gradient.

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Authors: Zeitoun D. G., Thierry Dana-Picard

Abstract:

The study of a real function of two real variables can be supported by visualization using a Computer Algebra System (CAS). One type of constraints of the system is due to the algorithms implemented, yielding continuous approximations of the given function by interpolation. This often masks discontinuities of the function and can provide strange plots, not compatible with the mathematics. In recent years, point based geometry has gained increasing attention as an alternative surface representation, both for efficient rendering and for flexible geometry processing of complex surfaces. In this paper we present different artifacts created by mesh surfaces near discontinuities and propose a point based method that controls and reduces these artifacts. A least squares penalty method for an automatic generation of the mesh that controls the behavior of the chosen function is presented. The special feature of this method is the ability to improve the accuracy of the surface visualization near a set of interior points where the function may be discontinuous. The present method is formulated as a minimax problem and the non uniform mesh is generated using an iterative algorithm. Results show that for large poorly conditioned matrices, the new algorithm gives more accurate results than the classical preconditioned conjugate algorithm.

Keywords: Function singularities, mesh generation, point allocation, visualization, collocation least squares method, Augmented Lagrangian method, Uzawa's Algorithm, Preconditioned Conjugate Gradien

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Authors: Guangbin Wang, Deyu Sun, Fuping Tan

Abstract:

In this paper, we present new preconditioned modified accelerated overrelaxation (MAOR) method for solving linear systems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned MAOR method converges faster than the MAOR method whenever the MAOR method is convergent. Finally, we give one numerical example to confirm our theoretical results.

Keywords: preconditioned, MAOR method, linear system, convergence, comparison.

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Authors: Guangbin Wang, Fuping Tan, Deyu Sun

Abstract:

In this paper, we present new preconditioned generalized mixed-type splitting (GMTS) methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GMTS methods converge faster than the GMTS method whenever the GMTS method is convergent. Finally, we give a numerical example to confirm our theoretical results.

Keywords: Preconditioned, GMTS method, linear system, convergence, comparison.

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Authors: Deyu Sun, Guangbin Wang

Abstract:

In this paper, we present preconditioned generalized accelerated overrelaxation (GAOR) methods for solving certain nonsingular linear system. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we give two numerical examples to confirm our theoretical results.

Keywords: Preconditioned, GAOR method, linear system, convergence, comparison.

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Authors: Guangbin Wang, Ning Zhang, Fuping Tan

Abstract:

In this paper, we present a preconditioned AOR-type iterative method for solving the linear systems Ax = b, where A is a Z-matrix. And give some comparison theorems to show that the rate of convergence of the preconditioned AOR-type iterative method is faster than the rate of convergence of the AOR-type iterative method.

Keywords: Z-matrix, AOR-type iterative method, precondition, comparison.

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Authors: Ahmad Alhawarat, Mustafa Mamat, Mohd Rivaie, Ismail Mohd

Abstract:

Conjugate gradient method has been enormously used to solve large scale unconstrained optimization problems due to the number of iteration, memory, CPU time, and convergence property, in this paper we find a new class of nonlinear conjugate gradient coefficient with global convergence properties proved by exact line search. The numerical results for our new βK give a good result when it compared with well known formulas.

Keywords: Conjugate gradient method, conjugate gradient coefficient, global convergence.

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Authors: Mansour A. Al Qubeissi

Abstract:

The current paper presents a numerical approach in solving the conjugate heat transfer problems. A heat conduction code is coupled internally with a computational fluid dynamics solver for developing a couple conjugate heat transfer solver. Methodology of treating non-matching meshes at interface has also been proposed. The validation results of 1D and 2D cases for the developed conjugate heat transfer code have shown close agreement with the solutions given by analysis.

Keywords: Computational Fluid Dynamics, Conjugate Heat transfer, Heat Conduction, Heat Transfer

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Authors: Lina Yan, Shiheng Wang, Ke Wang

Abstract:

A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

Keywords: preconditioning, M-matrix, Jacobi method, fuzzy linear system (FLS).

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Authors: Li Jiang, Baoguang Tian

Abstract:

In this paper, we present the preconditioned mixed-type splitting iterative method for solving the linear systems, Ax = b, where A is a Z-matrix. And we give some comparison theorems to show that the convergence rate of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give a numerical example to illustrate our results.

Keywords: Z-matrix, mixed-type splitting iterative method, precondition, comparison theorem, linear system.

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Authors: W. Suparta, M. A. Mohd. Ali, M. S. Jit Singh, B. Yatim, T. Motoba, N. Sato, A. Kadokura, G. Bjornsson

Abstract:

This paper observed the interhemispheric conjugate auroral activity occurred on 21 September 2009. The GPS derived ionospheric total electron content (TEC) during a weak substorm interval recorded at interhemispheric conjugate points at Husafell in Iceland and Syowa in Antarctica is investigated to look at their signatures on the auroral features. Selection of all-sky camera (ASC) images and keogram at Tjörnes and Syowa during the interval 00:47:54 – 00:50:14 UT on 21 September 2009 found that the auroral activity had exerted their influence on the GPS TEC as a consequence of varying interplanetary magnetic field (IMF) By polarity.

Keywords: Auroral activity, GPS TEC, Interhemispheric conjugate points, Responses

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Authors: Minghui Wang, Luping Xu, Juntao Zhang

Abstract:

Based on the conjugate gradient (CG) algorithm, the constrained matrix equation AXB=C and the associate optimal approximation problem are considered for the symmetric arrowhead matrix solutions in the premise of consistency. The convergence results of the method are presented. At last, a numerical example is given to illustrate the efficiency of this method.

Keywords: Iterative method, symmetric arrowhead matrix, conjugate gradient algorithm.

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Authors: N. M. Nawi, R. S. Ransing, M. R. Ransing

Abstract:

The conjugate gradient optimization algorithm is combined with the modified back propagation algorithm to yield a computationally efficient algorithm for training multilayer perceptron (MLP) networks (CGFR/AG). The computational efficiency is enhanced by adaptively modifying initial search direction as described in the following steps: (1) Modification on standard back propagation algorithm by introducing a gain variation term in the activation function, (2) Calculation of the gradient descent of error with respect to the weights and gains values and (3) the determination of a new search direction by using information calculated in step (2). The performance of the proposed method is demonstrated by comparing accuracy and computation time with the conjugate gradient algorithm used in MATLAB neural network toolbox. The results show that the computational efficiency of the proposed method was better than the standard conjugate gradient algorithm.

Keywords: Adaptive gain variation, back-propagation, activation function, conjugate gradient, search direction.

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Authors: Jamal Fathi Abu Hasna

Abstract:

There are two common methodologies to verify signatures: the functional approach and the parametric approach. This paper presents a new approach for dynamic handwritten signature verification (HSV) using the Neural Network with verification by the Conjugate Gradient Neural Network (NN). It is yet another avenue in the approach to HSV that is found to produce excellent results when compared with other methods of dynamic. Experimental results show the system is insensitive to the order of base-classifiers and gets a high verification ratio.

Keywords: Signature Verification, MATLAB Software, Conjugate Gradient, Segmentation, Skilled Forgery, and Genuine.

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Authors: V. Ghadamyari, F. Samadi, F. Kowsary

Abstract:

An inverse geometry problem is solved to predict an unknown irregular boundary profile. The aim is to minimize the objective function, which is the difference between real and computed temperatures, using three different versions of Conjugate Gradient Method. The gradient of the objective function, considered necessary in this method, obtained as a result of solving the adjoint equation. The abilities of three versions of Conjugate Gradient Method in predicting the boundary profile are compared using a numerical algorithm based on the method. The predicted shapes show that due to its convergence rate and accuracy of predicted values, the Powell-Beale version of the method is more effective than the Fletcher-Reeves and Polak –Ribiere versions.

Keywords: Boundary elements, Conjugate Gradient Method, Inverse Geometry Problem, Sensitivity equation.

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Authors: H. D. Ibrahim, H. C. Chinwenyi, H. N. Ude

Abstract:

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.

Keywords: conjugate gradient, linear equations, symmetric and positive definite matrix, Gauss-Seidel, Jacobi, algorithm

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Authors: E. N. Mathioudakis, E. P. Papadopoulou

Abstract:

The problem addressed herein is the efficient management of the Grid/Cluster intense computation involved, when the preconditioned Bi-CGSTAB Krylov method is employed for the iterative solution of the large and sparse linear system arising from the discretization of the Modified Helmholtz-Dirichlet problem by the Hermite Collocation method. Taking advantage of the Collocation ma-trix's red-black ordered structure we organize efficiently the whole computation and map it on a pipeline architecture with master-slave communication. Implementation, through MPI programming tools, is realized on a SUN V240 cluster, inter-connected through a 100Mbps and 1Gbps ethernet network,and its performance is presented by speedup measurements included.

Keywords: Collocation, Preconditioned Bi-CGSTAB, MPI, Grid and DSM Systems.

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Authors: Kai-Long Hsiao, BorMing Lee

Abstract:

In this study, an analysis has been performed for conjugate heat and mass transfer of a steady laminar boundary-layer mixed convection of magnetic hydrodynamic (MHD) flow with radiation effect of second grade subject to suction past a stretching sheet. Parameters E Nr, Gr, Gc, Ec and Sc represent the dominance of the viscoelastic fluid heat and mass transfer effect which have presented in governing equations, respectively. The similar transformation and the finite-difference method have been used to analyze the present problem. The conjugate heat and mass transfer results show that the non-Newtonian viscoelastic fluid has a better heat transfer effect than the Newtonian fluid. The free convection with a larger r G or c G has a good heat transfer effect better than a smaller r G or c G , and the radiative convection has a good heat transfer effect better than non-radiative convection.

Keywords: Conjugate heat and mass transfer, Radiation effect, Magnetic effect, Viscoelastic fluid, Viscous dissipation, Stretchingsheet.

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Authors: Zulhadi Zakaria, Nor Ashidi Mat Isa, Shahrel A. Suandi

Abstract:

This paper reports the study results on neural network training algorithm of numerical optimization techniques multiface detection in static images. The training algorithms involved are scale gradient conjugate backpropagation, conjugate gradient backpropagation with Polak-Riebre updates, conjugate gradient backpropagation with Fletcher-Reeves updates, one secant backpropagation and resilent backpropagation. The final result of each training algorithms for multiface detection application will also be discussed and compared.

Keywords: training algorithm, multiface, static image, neural network

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Authors: Ishak Hashim, Ammar Alsabery

Abstract:

The problem of conjugate free convection in a square cavity filled with nanofluid and heated from below by spatial wall temperature is studied numerically using the finite difference method. Water-based nanofluid with copper nanoparticles are chosen for the investigation. Governing equations are solved over a wide range of nanoparticle volume fraction (0 ≤ φ ≤ 0.2), wave number ((0 ≤ λ ≤ 4) and thermal conductivity ratio (0.44 ≤ Kr ≤ 6). The results presented for values of the governing parameters in terms of streamlines, isotherms and average Nusselt number. It is found that the flow behavior and the heat distribution are clearly enhanced with the increment of the non-uniform heating.

Keywords: Conjugate free convection, nanofluid, spatial temperature.

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Authors: Kai-Long Hsiao

Abstract:

A conjugate heat transfer for steady two-dimensional mixed convection with magnetic hydrodynamic (MHD) flow of an incompressible quiescent fluid over an unsteady thermal forming stretching sheet has been studied. A parameter, M, which is used to represent the dominance of the magnetic effect has been presented in governing equations. The similar transformation and an implicit finite-difference method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles, the wall unknown values of f''(0) and '(θ (0) for calculating the heat transfer of the similar boundary-layer flow are carried out as functions of the unsteadiness parameter (S), the Prandtl number (Pr), the space-dependent parameter (A) and temperature-dependent parameter (B) for heat source/sink and the magnetic parameter (M). The effects of these parameters have also discussed. At the results, it will produce greater heat transfer effect with a larger Pr and M, S, A, B will reduce heat transfer effects. At last, conjugate heat transfer for the free convection with a larger G has a good heat transfer effect better than a smaller G=0.

Keywords: Finite-difference method, Conjugate heat transfer, Unsteady Stretching Sheet, MHD, Mixed convection.

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Authors: N. M. Nawi, M. R. Ransing, R. S. Ransing

Abstract:

The conjugate gradient optimization algorithm usually used for nonlinear least squares is presented and is combined with the modified back propagation algorithm yielding a new fast training multilayer perceptron (MLP) algorithm (CGFR/AG). The approaches presented in the paper consist of three steps: (1) Modification on standard back propagation algorithm by introducing gain variation term of the activation function, (2) Calculating the gradient descent on error with respect to the weights and gains values and (3) the determination of the new search direction by exploiting the information calculated by gradient descent in step (2) as well as the previous search direction. The proposed method improved the training efficiency of back propagation algorithm by adaptively modifying the initial search direction. Performance of the proposed method is demonstrated by comparing to the conjugate gradient algorithm from neural network toolbox for the chosen benchmark. The results show that the number of iterations required by the proposed method to converge is less than 20% of what is required by the standard conjugate gradient and neural network toolbox algorithm.

Keywords: Back-propagation, activation function, conjugategradient, search direction, gain variation.

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Authors: Sergey Kopysov, Nikita Nedozhogin, Leonid Tonkov

Abstract:

The article presents a parallel iterative solver for large sparse linear systems which can be used on a heterogeneous platform. Traditionally, the problem of solving linear systems do not scale well on cluster containing multiple Central Processing Units (multi-CPUs cluster) or cluster containing multiple Graphics Processing Units (multi-GPUs cluster). For example, most of the attempts to implement the classical conjugate gradient method were at best counted in the same amount of time as the problem was enlarged. The paper proposes the pipelined variant of the conjugate gradient method (PCG), a formulation that is potentially better suited for hybrid CPU/GPU computing since it requires only one synchronization point per one iteration, instead of two for standard CG (Conjugate Gradient). The standard and pipelined CG methods need the vector entries generated by current GPU and other GPUs for matrix-vector product. So the communication between GPUs becomes a major performance bottleneck on miltiGPU cluster. The article presents an approach to minimize the communications between parallel parts of algorithms. Additionally, computation and communication can be overlapped to reduce the impact of data exchange. Using pipelined version of the CG method with one synchronization point, the possibility of asynchronous calculations and communications, load balancing between the CPU and GPU for solving the large linear systems allows for scalability. The algorithm is implemented with the combined use of technologies: MPI, OpenMP and CUDA. We show that almost optimum speed up on 8-CPU/2GPU may be reached (relatively to a one GPU execution). The parallelized solver achieves a speedup of up to 5.49 times on 16 NVIDIA Tesla GPUs, as compared to one GPU.

Keywords: Conjugate Gradient, GPU, parallel programming, pipelined algorithm.

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Authors: Xin-Lei Feng, Ting-Zhu Huang

Abstract:

According to conjugate gradient algorithm, a new consensus protocol algorithm of discrete-time multi-agent systems is presented, which can achieve finite-time consensus. Finally, a numerical example is given to illustrate our theoretical result.

Keywords: Consensus protocols; Graph theory; Multi-agent systems;Conjugate gradient algorithm; Finite-time.

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Authors: Habibis Saleh, Ishak Hashim

Abstract:

Conjugate natural convection in a differentially heated square enclosure containing a polygon shaped object is studied numerically in this article. The effect of various polygon types on the fluid flow and thermal performance of the enclosure is addressed for different thermal conductivities. The governing equations are modeled and solved numerically using the built-in finite element method of COMSOL software. It is found that the heat transfer rate remains stable by varying the polygon types.

Keywords: Natural convection, Polygon object, COMSOL

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Authors: Yue-Tzu Yang, Shu-Ching Liao

Abstract:

This study presents the numerical simulation of three-dimensional incompressible steady and laminar fluid flow and conjugate heat transfer of a trapezoidal microchannel heat sink using water as a cooling fluid in a silicon substrate. Navier-Stokes equations with conjugate energy equation are discretized by finite-volume method. We perform numerical computations for a range of 50 ≦ Re ≦ 600, 0.05W ≦ P ≦ 0.8W, 20W/cm² ≦q"≦ 40W/cm². The present study demonstrates the numerical optimization of a trapezoidal microchannel heat sink design using the response surface methodology (RSM) and the genetic algorithm method (GA). The results show that the average Nusselt number increases with an increase in the Reynolds number or pumping power, and the thermal resistance decreases as the pumping power increases. The thermal resistance of a trapezoidal microchannel is minimized for a constant heat flux and constant pumping power.

Keywords: Microchannel heat sinks, Conjugate heat transfer, Optimization, Genetic algorithm method.

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Authors: Yohei Saika, Hiroki Sakaematsu, Shota Akiyama

Abstract:

We investigated statistical performance of Bayesian inference using maximum entropy and MAP estimation for several models which approximated wave-fronts in remote sensing using SAR interferometry. Using Monte Carlo simulation for a set of wave-fronts generated by assumed true prior, we found that the method of maximum entropy realized the optimal performance around the Bayes-optimal conditions by using model of the true prior and the likelihood representing optical measurement due to the interferometer. Also, we found that the MAP estimation regarded as a deterministic limit of maximum entropy almost achieved the same performance as the Bayes-optimal solution for the set of wave-fronts. Then, we clarified that the MAP estimation perfectly carried out phase unwrapping without using prior information, and also that the MAP estimation realized accurate phase unwrapping using conjugate gradient (CG) method, if we assumed the model of the true prior appropriately.

Keywords: Bayesian inference using maximum entropy, MAP estimation using conjugate gradient method, SAR interferometry.

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Authors: Muzna Tariq, Ihtzaz Qamar

Abstract:

In most engineering cases, the working temperatures inside a combustion chamber are high enough that they lie beyond the operational range of thermocouples. Furthermore, design and manufacturing limitations restrict the use of internal thermocouples in many applications. Heat transfer inside a combustion chamber is caused due to interaction of the post-combustion hot fluid with the chamber wall. Heat transfer that involves an interaction between the fluid and solid is categorized as Conjugate Heat Transfer (CHT). Therefore, to satisfy the needs of CHT, CHT Analysis is performed by using ANSYS CFD tool to estimate theoretically precise thermocouple positions at the combustion chamber wall where excessive temperatures (beyond thermocouple range) can be avoided. In accordance with these Computational Fluid Dynamics (CFD) results, a combustion chamber is designed, and a prototype is manufactured with multiple thermocouple ports positioned at the specified distances so that the temperature of hot gases can be measured on the chamber wall where the temperatures do not exceed the thermocouple working range.

Keywords: Computational Fluid Dynamics, CFD, conduction, conjugate heat transfer, CHT, convection, fluid flow, thermocouples.

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Authors: Jatinderdeep Kaur

Abstract:

In this paper, the results of Kano from one dimensional cosine and sine series are extended to two dimensional cosine and sine series. To extend these results, some classes of coefficient sequences such as class of semi convexity and class R are extended from one dimension to two dimensions. Further, the function f(x, y) is two dimensional Fourier Cosine and Sine series or equivalently it represents an integrable function or not, has been studied. Moreover, some results are obtained which are generalization of Moricz’s results.

Keywords: Conjugate Dirichlet kernel, conjugate Fejer kernel, Fourier series, Semi-convexity.

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