关于解线性系统的一个有效的贝叶斯迭代方法

This paper concerns with the statistical methods for solving general linear systems. After a brief review of Bayesian perspective for inverse problems,a new and efficient iterative method for general linear systems from a Bayesian perspective is proposed.The convergence of this iterative method is proved,and the corresponding error analysis is studied.Finally, numerical experiments are given to support the efficiency of this iterative method,and some conclusions are obtained.     

Solving non-differentiable equations by a new one-point iterative method with memory

We construct a new iterative method for approximating the solutions of nonlinear operator equations, where the operator involved is not differentiable. The algorithm proposed does not need to evaluate derivatives and is more efficient than the secant method. For this, we extend a result of Traub for one-point iterative methods to one-point iterative methods with memory.     

Jacobi-Picard iteration method for the numerical solution of nonlinear initial value problems

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.     

一类Riccati矩阵方程广义自反解的双迭代算法

本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.     

对称正定矩阵的多级迭代法

本文主要研究解对称正定矩阵的多级迭代法,并对其收敛性进行证明。最后用数值实验验证此方法的有效性．多级迭代法特别适用于并行计算,并且可以被理解为古典迭代法的扩展,或共轭梯度法的预处理子。     

基于分数阶扩散方程的离散线性代数方程组迭代方法研究

本文构建一类双参数拟Toeplitz分裂（TQTS）迭代方法求解变系数非定常空间分数阶扩散方程.TQTS迭代法是基于QTS迭代法引入双参技术建立而成,通过选取适当的参数使迭代矩阵谱半径变得更小,从而有效提升收敛的速度.然后对TQTS迭代法进行收敛性分析,获得相应的收敛区域,并对迭代法中涉及的参数进行讨论,获得使迭代矩阵谱半径上界达到最小的最优参数的表达式.最后通过数值仿真实验验证TQTS迭代法的有效性,实验结果表明TQTS迭代法改进效果十分突出,在迭代时间和步数上均有明显的减小.     

On relaxed viscosity iterative methods for variational inequalities in Banach spaces

In this paper, we suggest and analyze a relaxed viscosity iterative method for a commutative family of nonexpansive self-mappings defined on a nonempty closed convex subset of a reflexive Banach space. We prove that the sequence of approximate solutions generated by the proposed iterative algorithm converges strongly to a solution of a variational inequality. Our relaxed viscosity iterative method is an extension and variant form of the original viscosity iterative method. The results of this paper can be viewed as an improvement and generalization of the previously known results that have appeared in the literature.     

用于求根问题的一个基于Thiele连分式的四阶收敛的迭代方法

利用截断的Thiele连分式,本文给出了一个求解非线性单变量方程的单步迭代方法,并证明了所提出的迭代方法具有四阶收敛性.最后,本文通过一些数值例子说明了所提出的方法的有效性和表现.     

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.