Banach空间中渐近非扩张映象的带误差的修正的Ishikawa与Mann迭代程序

设E是满足Opial条件的一致凸Banach空间,C是E的一非空闭凸子集,T:C→C是渐近非扩张映象.又设对任给的x1∈C,序列{xn}由下列带误差的修正的Ishikawa迭代程序生成:其中, 是C中的序列,使得 且数列 满足下列条件(i)和(ii)之一: (i)tn∈[a,b]且sn∈[O,b];(ii)tn∈[a,b]且sn∈[a,b],这里,常数a,b满足0相似文献   

求解大型超定线性代数方程组的块Kaczmarz算法

正1 引言考虑大型超定线性代数方程组Ax=b,(1)其中 A ∈ C~(m×n) (m n),b ∈C~m.当m=n时,线性代数方程组求解的相关理论和算法较为成熟,但在很多实际问题中,系数矩阵A的行数和列数不相等(m≠n),如超定或欠定线性代数方程组.因此,有必要研究此类线性代数方程组的数值解法.在结构分析,计算机辅助几何设计,图像恢复,模型参数估计等众多领域中,经常需要求解大型超定线性代数方程组.Vuik [1]研究了大型超定线性代数方程组最小二乘问题的预处理Krylov迭代方法;Bai [2]提出列分解松弛法;Yin[3]提出了求解大型稀疏最小二乘问题的不完备Givens正交化的预处理GMRES方法;Hayami[4]考虑引入一个新的矩阵将GMRES方法应用到最小二乘问题,求得方程组的最小二乘解;Finta [5]推导了加权超定线性代数方程组的梯度法,并证明该方法是收敛的.     

求解离散不适定问题的正则化GMERR方法

迭代极小残差方法是求解大型线性方程组的常用方法, 通常用残差范数控制迭代过程.但对于不适定问题, 即使残差范数下降, 误差范数未必下降. 对大型离散不适定问题,组合广义最小误差(GMERR)方法和截断奇异值分解(TSVD)正则化方法, 并利用广义交叉校验准则(GCV)确定正则化参数,提出了求解大型不适定问题的正则化GMERR方法.数值结果表明, 正则化GMERR方法优于正则化GMRES方法.     

求地震反演的I_1模极小化模型的解

1 引 言 在文[1]中提出了地震反演的l_1模极小化模型是: min ψ(x)=||x||1, (1.1) s.t. Ax=b,其中A∈R~(m×n),m相似文献   

关于Dini定理

关于函数序列或函数级数的一致收敛性判别准则 ,我们熟知的有 M判别法 ,Abel定理及Dirichlet定理 .在作者的《数学分析》(下册 P95 ,高等教育出版社 ,1 995年 )中还介绍了 Dini定理 ,以下称之为第一 Dini定理 .第一 Dini定理　设 [a,b] R是一有界闭区间 , n∈N,fn∶ [a,b]→R是一连续函数且满足下述条件 :1 )函数序列 { fn}是单调的 ,即 n∈ N ,fn≤ fn+ 1或 n∈ N ,fn≥ fn+ 1.2 )函数序列 { fn}在 [a,b]上逐点收敛于一连续函数 f :[a,b]→ R ,那末函数序列 { fn}在 [a,b]上一致收敛于函数 f.注意 ,上述条件 1 )中的单调性是指函数…     

加权逼近中的A最大类

设 C_[a,b]是定义在[a,b]上被赋予一致范数的实连续函数空间,G 是 C_[a,b]的一个真子集,ω是一固定的非负连续函数.如果 g∈G 使     

Banach空间中渐近非扩张映射的不动点迭代

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .     

一类可逆的对角占优矩阵与Jacobi迭代法

<正> 用Jacobi 迭代法解线性方程组AX=b(其中A∈R~(n×n)、b∈R~n.X∈R~n)时,一般假定A 为可逆阵且a_(ii)≠0(i=1,2,…n)。文[1]指出.如果矩阵A 为严格对角占优阵,则Ja obi 迭代过程是收敛的。‘严格对角占优’这个条件是比较强的,它限制了Jacobi 迭代法的应用范围。实际     

一类带Hardy-Sobolev临界指数的奇异Kirchhoff型方程正解的存在性

本文研究了一类带Hardy-Sobolev临界指数的奇异Kirchhoff型{-(a+b∫_Ω︱▽u︱~2dx)△u=u~(5-2s)/︱x︱~s+λu~(-γ),x∈Ω,u0,x∈Ω,u=0,x∈δΩ方程其中ΩR~3是一个有界开区域且具有光滑边界δΩ,0∈Ω,a,b≥0且a+b0,λ0,0γ1,0≤s1.利用变分方法,获得了该问题的一个正局部极小解,补充了文献[1]的结果.     

GMRES方法的收敛率

1 引 言 GMRES方法是目前求解大型稀疏非对称线性方程组 Ax=b,A∈R~(n×n);x,b∈R~n (1)最为流行的方法之一.设x~((0))是(1)解的初始估计,r~((0))=b-Ax~((0))是初始残量,K_k=span{r~((0)),Ar~((0)),…A~(k-1)r~((0))}为由r~((0))和A产生的Krylov子空间.GMRES方法的第k步     

一种新型压缩预处理CGS算法

CGS算法是求解大型非对称线性方程组的常用算法，然而该算法无极小残差性质，因此它常因出现较大的中间剩余向量而出现典型的不规则收敛行为．本根据IRA方法提出了一种压缩预处理CGS方法，数值实验表明这种算法在一定程度上减小了迭代算法在收敛过程中的剩余问题，从而使得算法具有更好的稳定性，该法构造简单，减少了收敛次数，加快了收敛速度．     

Krylov subspace methods with deflation and balancing preconditioners for least squares problems

For solving least squares problems, the CGLS method is a typical method in the point of view of iterative methods. When the least squares problems are ill-conditioned, the convergence behavior of the CGLS method will present a deteriorated result. We expect to select other iterative Krylov subspace methods to overcome the disadvantage of CGLS. Here the GMRES method is a suitable algorithm for the reason that it is derived from the minimal residual norm approach, which coincides with least squares problems. Ken Hayami proposed BAGMRES for solving least squares problems in [\emph{GMRES Methods for Least Squares Problems, SIAM J. Matrix Anal. Appl., 31(2010)}, pp.2400-2430]. The deflation and balancing preconditioners can optimize the convergence rate through modulating spectral distribution. Hence, in this paper we utilize preconditioned iterative Krylov subspace methods with deflation and balancing preconditioners in order to solve ill-conditioned least squares problems. Numerical experiments show that the methods proposed in this paper are better than the CGLS method.     

CHEBYSHEV WEIGHTED NORM LEAST-SQUARES SPECTRAL METHODS FOR THE ELLIPTIC PROBLEM

We develop and analyze a first-order system least-squares spectral method for the second-order elhptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares functional defined by the sum of the Lw^2- and Hw^-1- norm of the residual equations and then we eplace the negative norm by the discrete negative norm and analyze the discrete Chebyshev weighted least-squares method. The spectral convergence is derived for the proposed method. We also present various numerical experiments. The Legendre weighted least-squares method can be easily developed by following this paper.     

Efficient solutions schemes for interface problems

A computationally efficient two-level iterative scheme is proposed for the solution of the interface problems with Lagrange multipliers, where the oscillatory part of the solution is resolved by means off smoothing using a new, efficient preconditioner whereas the smooth component of the solution is captured by the collocation-based problem on the auxilliary grid, that is solved directly using a sparse direct solver. A simple adaptive feature is built into the proposed solution method in order to guarantee convergence for ill-conditioned problems. Nmerical results presented for example problems including that of a Boeing crown panel show that the proposed tww-level solution technique outperfrmsnce the standard, single level iterative and direect solvers.     

A derivative-free algorithm for systems of nonlinear inequalities

Recently a new derivative-free algorithm has been proposed for the solution of linearly constrained finite minimax problems. This derivative-free algorithm is based on a smoothing technique that allows one to take into account the non-smoothness of the max function. In this paper, we investigate, both from a theoretical and computational point of view, the behavior of the minmax algorithm when used to solve systems of nonlinear inequalities when derivatives are unavailable. In particular, we show an interesting property of the algorithm, namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locates a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. Moreover, we carried out numerical experimentation and comparison of the method against a standard pattern search minimization method. The obtained results confirm that the good theoretical properties of the method correspond to interesting numerical performance. Moreover, the algorithm compares favorably with a standard derivative-free method, and this seems to indicate that extending the smoothing technique to pattern search algorithms can be beneficial.     

An iterative method for solving semismooth equations

In this paper, we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations. At each iteration, a trust region subproblem is solved. The solution of the trust region subproblem provides a descent direction for the norm of a smoothing function. By using a backtracking line search, a steplength is determined. The proposed method shares advantages of trust region methods and line search methods. Under appropriate conditions, the proposed method is proved to be globally and superlinearly convergent. In particular, we show that after finitely many iterations, the unit step is always accepted and the method reduces to a smoothing Newton method.     

Generalizations and modifications of the GMRES iterative method

For solving nonsymmetric linear systems, the well-known GMRES method is considered to be a stable method; however, the work per iteration increases as the number of iterations increases. We consider two new iterative methods GGMRES and MGMRES, which are a generalization and a modification of the GMRES method, respectively. Instead of using a minimization condition as in the derivation of GGMRES, we use a Galerkin condition to derive the MGMRES method. We also introduce another new iterative method, LAN/MGMRES, which is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method. A computer program has been written based on the use of the LAN/MGMRES algorithm for solving nonsymmetric linear systems arising from certain elliptic problems. Numerical tests are presented comparing this algorithm with some other commonly used iterative algorithms. These preliminary tests of the LAN/MGMRES algorithm show that it is comparable in terms of both the approximate number of iterations and the overall convergence behavior. This revised version was published online in June 2006 with corrections to the Cover Date.     

2n阶Lidstone边值问题正解的逐次迭代

研究了2n阶Lidstone边值问题正解的逐次迭代,其中非线性项依赖于所有偶数阶导数.通过考察非线性项在某些有国介集合上的“高度”并利用单调迭代方法构造了一个逐次迭代程序.这个迭代程序从一个多项式开始并且是可行的.使用这个结论获得了m个正解的迭代方法,其中m是一个任意的自然数.     

A generalization of parameterized inexact Uzawa method for generalized saddle point problems

Recently, a class of parameterized inexact Uzawa methods has been proposed for generalized saddle point problems by Bai and Wang [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932], and a generalization of the inexact parameterized Uzawa method has been studied for augmented linear systems by Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. (2008)]. This paper is concerned about a generalization of the parameterized inexact Uzawa method for solving the generalized saddle point problems with nonzero (2, 2) blocks. Some new iterative methods are presented and their convergence are studied in depth. By choosing different parameter matrices, we derive a series of existing and new iterative methods, including the preconditioned Uzawa method, the inexact Uzawa method, the SOR-like method, the GSOR method, the GIAOR method, the PIU method, the APIU method and so on. Numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact Uzawa methods.     

Truncated Smoothing Newton Method for l_∞ Fitting Rotated Cones

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than l2 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large...