Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. They are now mostly used as precondition-ers for the popular iterative solvers. In this paper a generalization of these methods are proposed and their convergence properties are studied. Some numerical experiments are given to show the efficiency of the new methods.     

CONVERGENCE ANALYSIS ON ITERATIVE METHODS FOR SEMIDEFINITE SYSTEMS

The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sumcient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regradled as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8, 9].     

Spectral Collocation Method for Caputo Fractional Multi-point Value Problems北大核心CSCD

Spectral collocation method is investigated for the nonlinear Caputo fractional multi-point value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular Volterra–Fredholm integral equations obtained from the nonlinear Caputo fractional multi-point value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with Volterra–Fredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multi-point value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects. © 2022 Chinese Academy of Sciences. All rights reserved.     

MEAN CONVERGENCE OF HERMITE-FEJ(E)R TYPE INTERPOLATION ON AN ARBITRARY SYSTEM OF NODES

In this paper sufficient conditions for mean convergence and rate of convergence of Hermite-Fejer type interpolation in the Lp norm on an arbitrary system of nodes are presented.     

ON A FAMILY OF CHEBYSHEV-HALLEY TYPE METHODS IN BANACH SPACE UNDER WEAKER SMALE CONDITION

In this paper, we discuss local convergence of a family of Chebychev-Halley type methods with a parameter θ∈[0,1] in Banach space using Smale-type δ criterion under 2-th γ-condition. We will see that the properties of the condition used for local convergence is much more different from that used in [6][15] for the semi-local convergence.     

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

In this paper,the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated.Some sufficient conditions for the convergence are provided.In addition,the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained.The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

Two algorithms for two-phase Stefan type problems

In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.     

The Asymptotic Behavior for Numerical Solution of a Volterra Equation

Long-time asymptotic stability and convergence properties for the numerical solution of a Volterra equation of parabolic type are studied.The methods are based on the first-second order backward difference methods.The memory term is approximated by the comvolution quadrature and the interpolant quadrature.Discretization of the spatial partial differential operators by the finite element method is also considered.     

HIGH-ORDER NONLINEAR GALERKIN METHODS

The nonlinear Galerkin methods are numerical schemes well adapted to the long-term integration of nonlinear evolution partial differential equations. In this paper, a class of high-order nonlinear Galerkin methods are provided. Moreover, convergence results with high-order spectral accuracy are derived for the schemes introduced.     

SCHWARZ DOMAIN DECOMPOSITION ALGORITHMS FOR BOUNDARY ELEMENT METHOD

In this paper, we introduce two Schwarz type domain decomposition algorithms for solving boundary element equations, which decompose the original problem defined on global boundary surface into several ones defined on sub-domains so that they may be solved ileratively or parallelly. The convergence of these methods are also proved.     

GLOBALLY CONVERGENT INEXACT GENERALIZED NEWTON METHODS WITH DECREASING NORM OF THE GRADIENT

AbstractIn this paper, motivated by the Martinez and Qi methods[l], we propose one type of globally convergent inexact generalized Newton methods to solve unconstrained optimization problems in which the objective functions are not twice differentiable, but have LC gradient. They make the norm of the gradient decreasing. These methods are implementable and globally convergent. We prove that the algorithms have superlinear convergence rates under some mile conditions.The methods may also be used to solve nonsmooth equations.     

基于凸优函数的Newton类方法的收敛定理

本文研究了求解Banach空间上非线性算子方程f(x)=0的Newton类方法的收敛性.利用优函数原理,在A(x0)1f满足关于某一凸优函数的广义Lipschitz条件下,得到了Newton类方法的一个半局部收敛定理.同时,当f和A(x)及初始点x0给定时,针对广义Lipschitz条件构造了相应的优函数,推广了Newton类方法的相关结果.     

关于PDAOR算法的广义STEIN—ROSENBERG型定理

1．引言众所周知，许多微分方程（组）经过有限差分或有限元离散，均可归结为大型分块线性代数方程组的数值求解问题，这里n。（5。5N）为给定的N个正整数，满足Zn。＝n．为利用多处理机系统有效而准t’z＝1确地得到JI．n的近似解．诵过合理地分解系统〔1．1），并有机地运用加速超松弛技术，【11提出了一类新的求解大型分块线性代数方程组（1．1）的并行分解型加速超松弛迭代算法，即PDAOR－一算法．这类算法具有很强的并行功能和良好的数值性质．大量数值实验表明，较之经典的AOR算法，PDAOR－一算法具有更快的收敛速度，更大的收…     

非拟牛顿非凸族的收敛性

１．引言 对于无约束最优化问题拟牛顿法是目前最成熟,应用最广泛的解法之一．近二十多年来,对拟牛顿法收敛性质的研究一直是非线性最优化算法理论研究的热点．带非精确搜索的拟牛顿算法的研究是从１９７６年 Ｐｏｗｅｌｌ［１］开始,他证明了带 Ｗｏｌｆｅ搜索 ＢＦＧＳ算法的全局收敛性和超线性收敛性． １９７８年 Ｂｙｒｄ, Ｎｏｃｅｄａｌ; Ｙａ－Ｘｉａｎｇ Ｙｕａｎ［３］成功地将 Ｐｏｗｅｌｌ的结果推广到限制的 Ｂｒｏｓｄｅｎ凸族． １９８９年, Ｎｏｃｅｄａｌ［４］在目标函数一致凸的条件下,证明了带回追搜索的ＢＦＧ…     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.     

具有参数的不带有导数的平方收敛的迭代法

1.引 言 考虑数值求解非线性方程 f(x)=0, (1)其中实值函数f(x)在实零点x*的某邻域U(x*)内连续可微且f'(x)≠0. 牛顿法是科学与工程计算中数值求解(1)的常用数值方法.虽然它一般至少是二阶收敛的,但它需要调用导数值,这使其应用受到限制.我们修改牛顿法,用割线代替切线可得不带     

GAOR迭代法的收敛性

当A为实对称矩阵时,[1]中在D_i选取较特殊的条件下,证明了GAOR迭代法收敛的充要条件为A是正定矩阵. 设A为Hermite矩阵,进一步讨论GAOR迭代法收敛的充要条件. 以下记 B=D_1~(-1)(C_L+C_U).     

非线性互补问题的两种数值解法

本文研究了非线性互补问题的两类数值求解方法.在经典LQP算法及LevenbergMarquardt算法的基础上,构造了两种新算法,并证明了这两种新算法的收敛性.数值实验表明,新算法对测试问题优于已有算法.     

THE APPROXIMATE COMPUTATION OF HYPERSINGULAR INTEGRALS ON INTERVAL

In this paper some numerical methods for computing hypersingular integrals on interval are given. Using geometric meshes, these methods lead 10 an exponential convergence in the range of engineering compulation. A numerical example shows their effeclivity and accuracy.     

ON THE CONVERGENCE OF KING-WERNER ITERATION METHOD IN BANACH SPACE ON THE CONVERGENCE OF KING-WERNER ITERATION METHOD IN BANACH SPACE

1. IntroductionLetf(x) = 0 (1.1)where j: X - Y is a norilineax operator which maps Banach space X into Baaedspace Y. The well-known iteration methods for solving (1.1) are the Nixon methodaam very ldnds of its improvement methods. One of them is the so called King-Wernerm6thod denned bykw(p, x03 yo):which is established by King in [7], Werner in [12] in d~nt formulas, respectively.It is interesting that the method (1.2) is of order 1 fi with the same functioncompotatinn coSt and twO ti…