Spectral Radius Of Iteration Matrix Research Articles

Convergence analysis of Lattice Boltzmann method for Stokes flow in digital rock characterization

The absolute permeability is a key parameter for characterization of reservoir rocks. It can be estimated numerically by combining digital rocks and Stokes flow simulation using Lattice Boltzmann Method (LBM). In previous research, the LBM is usually implemented as an iterative process, which continues until the changes in estimates of the parameters between two consecutive iterations reach below a certain threshold level. We remark that this termination criterion is not appropriate and may lead to low accuracy in the simulation results. Here we analyze the convergence of LBM in various tests including the Poiseuille flow between two parallel plates, and several types of digital rocks (dune sand, sandstone and carbonates). The logarithm of relative error of each estimate compared with the estimate at infinite time (as stable state) is found to have a linear relation with regard to the number of iterations, indicating an exponential rate of convergence. If the difference of errors between two consecutive iterations is used as the termination condition, the result could be far from a stable state. Instead, we recommend using the decay trend of the error difference in the LBM simulation to provide an accurate termination criterion, which serves as a practical and more adequate way for characterization of reservoir rocks. We also provided the theoretical explanation of convergence rate, which is related to the spectral radius of iterative matrix in linear algebra.

A new approximation algorithm for solving generalized Lyapunov matrix equations

In this paper, we propose a new approximation algorithm for solving generalized Lyapunov matrix equations. We also present a convergence analysis for this algorithm. In each step of this algorithm two standard Lyapunov matrix equations with real coefficient matrices should be solved. Then we determine the optimal parameter to minimize the corresponding spectral radius of iteration matrix to obtain fastest speed of convergence. Finally some numerical examples are given to prove the capability of the present algorithm and a comparison is made with the existing results.

Two lopsided TSCSP (LTSCSP) iteration methods for solution of complex symmetric positive definite linear systems

In this paper we make two new lopsided methods (LTSCSP1 and LTSCSP2) based on the two-scale-splitting (TSCSP) method and show that the convergence speed of TSCSP iteration method can be increased under some conditions without adding any additional parameter. The convergence analysis of the new methods in detail is given. Then we will obtain the quasi-optimal parameter to minimize the spectral radius of iteration matrix for the new methods. The inexact version of these methods is derived. To illustrate the effectiveness of the proposed framework, several numerical examples are given.

Solving complex Sylvester matrix equation by accelerated double-step scale splitting (ADSS) method

Accelerated double-step scale splitting (ADSS) approach is an efficient and fast method for solving a class of large complex symmetric linear equations which has been recently presented by parameterized DSS method. In this paper, we will apply ADSS scheme for solving complex Sylvester matrix equation. It will be proved analytically that the ADSS iteration method is faster than the DSS iteration method. Moreover, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and results will be reported to support the theoretical claims.

Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems

In this paper, we introduce and study an accelerated double-step scale splitting (ADSS) iteration method for solving complex linear systems. The convergence of the ADSS iteration method is determined under suitable conditions. Also each iteration of ADSS method requires the solution of two linear systems that their coefficient matrices are real symmetric positive definite. We analytically prove that the ADSS iteration method is faster than the DSS iteration method. Moreover, to increase the convergence rate of this method, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and simulation results will be reported to support the theoretical results.

A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation

In this study, based on the MHSS (Modified Hermitian and skew-Hermitian splitting) method, we will present a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices. The new method (GMHSS) is a four-parameter iteration procedure where the iterative sequence is unconditionally convergent to the unique solution of the Sylvester equation. Then to improve the GMHSS method, the inexact version of the GMHSS iterative method (IGMHSS) will be described and will be analyzed. Also, by using a new idea, we try to minimize the upper bound of the spectral radius of iteration matrix. Two test problems are given to illustrate the efficiency of the new approach.

A special generalized HSS method for discrete ill-posed problems

In this paper, we use a new special generalized Hermitian and skew-Hermitian splitting (SGHSS) method for solving ill-posed inverse problems. Based on an augmented system formulation, we apply a new splitting for the Hermitian part of the coefficient matrix and discuss on the convergence of this iterative method. Moreover, the optimal parameter minimizing the spectral radius of iteration matrix is derived. Finally, we present some numerical examples to demonstrate the performance of the presented method.

On HSS-based sequential two-stage method for non-Hermitian saddle point problems

For large sparse saddle point problems with symmetric positive definite (1,1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1,1)-block of the preconditioned non-Hermitian saddle point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian saddle point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.

Preconditioned modified AOR method for systems of linear equations

In this paper, we present three kinds of preconditioners for preconditioned modified accelerated over-relaxation (PMAOR) method to solve systems of linear equations. We show that the spectral radius of iteration matrix for the PMAOR method is smaller than that for modified accelerated over-relaxation (MAOR) method including their comparison. The comparison results show that the convergence rate of the PMAOR method is better than the rate of the MAOR method. We provide some numerical results for the evidence of our method. Also we compare our numerical results with solutions by variational iteration method. Copyright © 2009 John Wiley & Sons, Ltd.

Solving Large Scale Unconstrained Minimization Problems by a New ODE Numerical Integration Method

In reference [1], for large scale nonlinear equations , a new ODE solving method was given. This paper is a continuous work. Here has gradient structure i.e. , is a scalar function. The eigenvalues of the Jacobian of ; or the Hessian of , are all real number. So the new method is very suitable for this structure. For quadratic function the convergence was proved and the spectral radius of iteration matrix was given and compared with traditional method. Examples show for large scale problems (dimension ) the new method is very efficient.

Symmetric modified AOR method to solve systems of linear equations

We propose a class of symmetric modified accelerated overrelaxation (SMAOR) methods for solving large sparse linear systems. The convergence region of the method has been investigated. Numerical examples indicate that the SMAOR method is better than other methods such as accelerated overrelaxation(AOR) and modified accelerated overrelaxation(MAOR) methods, since the spectral radius of iteration matrix in SMAOR method is less than that of the other methods. Also, we apply the method to solve a real boundary value problem.

Some results on a generalized alternating iterative method

Climent and Perea [J.-J. Climent, C. Perea, Convergence and comparison theorems for a generalized alternating iterative method, Appl. Math. Comput. 143 (2003) 1–14] introduced a generalized alternating iterative method. In this paper, we establish convergence results for a nonsingular H-matrix, upper bound to the spectral radius of iterative matrix and comparison theorem for a monotone matrix. Moreover, we give some numerical examples to show our results.

The optimal preconditioner of strictly diagonally dominant Z-matrix

In this paper, we present a series of new preconditioners with parameters of strictly diagonally dominant Z-matrix, which contain properly two kinds of known preconditioners as special cases. Moreover, we prove the monotonicity of spectral radiuses of iterative matrices with respect to the parameters and some comparison theorems. The results obtained show that the bigger the parameter k is(i.e., we select the more upper right diagonal elements to be the preconditioner), the less the spectral radius of iterative matrix is. A numerical example generated randomly is provided to illustrate the theoretical results.

Nouvelle approche pour la discrétisation de flux diffusifs en volumes finis à forte obliquité

We propose a new approach for the discretization of diffusive flux on a non orthogonal mesh based on the Deferred correction introduced by Peric. This new method is applied successfully to the solution of a Poisson problem in quadrangular domains meshed with very distorted control volumes. The interest of this approach lies in the fact that the precision of the used schemes is conserved despite meshes distortion level. The comparative study of the value of the spectral radius of iteration matrix enables us to explain why our Deferred correction approach is better than that of Peric, which diverges on highly skewed meshes. To cite this article: Y.M. Ahipo, Ph. Traore, C. R. Mecanique 334 (2006).