Accelerated Overrelaxation Method Research Articles

Accelerated Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

In this paper, by applying the accelerated technique and relaxation two-sweep strategy to the modulus-based matrix splitting iteration method, we establish the accelerated relaxation two-sweep modulus-based matrix splitting method. The proposed method contains the accelerated modulus-based matrix splitting and the generalized accelerated modulus-based matrix splitting methods presented recently as special cases. Compared with some existing methods, the proposed method can use more information during each iteration, which may lead to higher computational efficiency. We investigate the convergence properties of the new method, and prove that it is convergent under certain assumptions. Also, for the accelerated relaxation two-sweep modulus-based accelerated overrelaxation method, we analyze the convergence regions of the parameters in detail. Numerical examples are offered to show that the proposed method is efficient and performs better than the generalized accelerated modulus-based matrix splitting one in actual implementation.

The hybird methods of projection-splitting for solving tensor split feasibility problem

<abstract><p>Projection-type methods are studied widely and deeply to solve multiples-sets split feasibility problem. From the perspective of splitting iteration, in this paper, the efficient tensor projection-splitting iteration methods are firstly investigated for solving tensor split feasibility problem, which is properly transformed into a tensor equation by taking advantage of projection operator. Then, accelerated overrelaxation method and symmetric (alternating) accelerated overrelaxation method are further generalized to solve this problem from general system of linear equations to multi-linear systems. Theoretically, the convergence is proven by the analysis of spectral radius of splitting iteration tensor. Numerical experiments demonstrate the efficiency of the presented method.</p></abstract>

Efficient Splitting Methods for Solving Tensor Absolute Value Equation

The tensor absolute value equation is a class of interesting structured multilinear systems. In this article, from the perspective of pure numerical algebra, we first consider a tensor-type successive over-relaxation method (SOR) (called TSOR) and tensor-type accelerated over-relaxation method (AOR) (called TAOR) for solving tensor absolute value equations. Furthermore, one type of preconditioned tensor splitting method is also applied for solving structured multilinear systems. Numerical experiments adequately demonstrate the efficiency of the presented methods.

Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors

The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time.

The solution of the absolute value equations using two generalized accelerated overrelaxation methods

Finding the solution of the absolute value equations (AVEs) has attracted much attention in recent years. In this paper, we propose and analyze two generalized accelerated overrelaxation (AOR) methods for solving AVEs [Formula: see text], where [Formula: see text] is an [Formula: see text]-matrix. Furthermore, we discuss the convergence of the methods under some suitable assumptions. Numerical results are given to verify the effectiveness of our methods.

Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.

Novel accelerated methods of tensor splitting iteration for solving multi-systems

Tensor splitting iteration method is a class of popular technique for solving multi-linear systems. In this paper, we present one kind of efficient alternating splitting iteration method, and further generalize accelerated overrelaxation method (AOR) and symmetric accelerated overrelaxation method (SAOR) from linear systems to multi-systems. Then, one type of preconditioned (alternating) tensor splitting method is also applied for solving multi-systems. Numerical experiments illustrate the efficiency of the provided methods.

Investigation of Triangle Element Analysis for the Solutions of 2D Poisson Equations via AOR method

In earlier studies of iterative approaches, the accelerated over relaxation (AOR) method has been pointed out to be much faster as compared to the existing successive over re- laxation (SOR) and Gauss Seidel (GS) methods. Due to the effectiveness of this method, the foremost goal of this paper is to demonstrate the use of the AOR method together with triangle element solutions based on the Galerkin scheme method. The effectiveness of this method has been shown via results of numerical experiments, which are logged and examined. The findings show that the AOR method is superior as compared with the existing SOR and GS methods.

Preconditioned Generalized Accelerated Overrelaxation Methods for Solving Certain Nonsingular Linear System

Preconditioned Generalized Accelerated Overrelaxation Methods for Solving Certain Nonsingular Linear System

On the Accelerated Overrelaxation Method

A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.

On the Solution of the Linear Complementarity Problem by the Generalized Accelerated Overrelaxation Iterative Method

In the present work, we determine intervals of convergence for the various parameters involved for what is known as the generalized accelerated overrelaxation iterative method for the solution of the linear complementarity problem. The convergence intervals found constitute sufficient conditions for the generalized accelerated overrelaxation method to converge and are better than what have been known so far.

Convergence Analysis of Modified Iterative Methods to Solve Linear Systems

In the past years, a growing interest to solve linear systems with modified iterative methods has been shown by researchers. Recently, Dehghan and Hajarian (J Vib Control, doi: 10.1177/10775463124 , 2012)based on preconditioned methods, introduced modified accelerated overrelaxation methods for solving linear systems. These authors stated that the best property of the mentioned methods is that they can be used under mild conditions than the Milaszewicz’s (Linear Algebra Appl 93:161–170, 1987) preconditioner. In this paper, we show that the Milaszewicz’s preconditioner is applicable under mild conditions and also, under these conditions, Milaszewicz’s preconditioner is superior to the Dehghan and Hajarian’s preconditioner. Numerical examples are also given to illustrate our results.

Implicit finite difference solution for time-fractional diffusion equations using AOR method

In this paper, we derive an implicit finite difference approximation equation of the one-dimensional linear time fractional diffusion equations, based on the Caputo's time fractional derivative. Then this approximation equation leads the corresponding system of linear equation, which is large scale and sparse. Due to the characteristics of the coefficient matrix, we use the Accelerated Over-Relaxation (AOR) iterative method for solving the generated linear system. One example of the problem is presented to illustrate the effectiveness of AOR method. The numerical results of this study show that the proposed iterative method is superior compared with the existing one weighted parameter iterative method.

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some Results on Preconditioned Modified Accelerated Overrelaxation Method

Some Results on Preconditioned Modified Accelerated Overrelaxation Method

A New Version of the Accelerated Overrelaxation Iterative Method

Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant,L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant,H-matrices, symmetric positive definite matrices, andL-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method.

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method