偏序下的区间松弛法

§1. 引言 本文给出了求解非线性方程组 f(x)=0,f:D?R~n→R~m (1.1)在偏序下的区间松弛法,它是在[1]的基础上将区间迭代与Newton-SOR 迭代结合得到的一种便于计算且收敛较快的序区间N-SOR松弛法,也是单调N-SOR迭代法的推广.§2给出了偏序下的区间Krawczyk算子,它是区间 Newton算子的推广,同样具     

求解奇异问题一类新的加速迭代法

格式(1.1)每步只需求一次导算子的逆,计算量比现有的加速迭代格式均少,同时具有高阶收敛性。格式(1.2)与文[1]中提出的迭代格式相比,计算量基本相同,但其收敛速度却较快。我们在§2中给出算法(1.1)和(1.2)的收敛性定理及误差估计。对于高阶奇异问题,§3中也给出了相应的加速迭代格式和收敛性定理。§4中给出数值例子。     

分裂映射的两侧逼近区间割线法

对于求解非线性方程组的区间迭代法,若利用Moore检验,可判断解的存在唯一性.[2]中在偏序下给出的区间Newton型方法,也有同样特性,本文利用f:D?R~n→R~n的斜度构造的区间割线算子,也可用于检验方程组解的存在唯一性,但它不用计算f的导数,针对f的不同分裂,还可以构造不同的两侧逼近割线法.分裂得当,便于求逆,使计算     

算子变形的一种方法

本文提出算子的一种变形方法,可以改善算子的逼近度.§1叙述这种变形的方法,§2讨论它的某些应用.§1.算子变形的一种方法C~n[α,b]表示区间[α,b]上具有 n 阶连续导数的函数类,用 C[α,b]表示[α,b]上的连续函数全体.有线性算子     

关于复方程组的圆盘迭代法

§1.导言 关于非线性复方程组: f(z)=0,f:G?C~n→C~n的圆盘迭代,[1]中曾考虑过圆盘Newton法,它需要计算圆盘逆阵,因此计算最大.其中还给出了一种Krawczyk-Moore型算法,本文的目的就是对这一结果作进一步改进,     

二重福里哀级数系数的性质

<正> §1.引言.设 f(x)为以2π为周期的周期函数,其福里哀展开式为下列各事是大家熟知的:设 f(x)在一个基本区间(0,2π)不有界变差的函数,则     

在点估计判据下Euler级数、Euler迭代族以及Hauey迭代族的收敛性

论文证明了,当 S.Smale[1—3]的点估计判据α(f,z)=‖Df(z)~-1f(z)‖·(?)‖Df(z)~(-1)D~nf(z)/n!‖~(1/(n-1))≤3-22~(1/2)时,求 Banach 空间解析映照f零点ζ的 Newton 迭代的两族高阶推广以及ζ的逆级数都收敛,并且对其中每一个极限来说,条件中的常数3-22~(1/2)都是最好可能的.对其中以f在z的[1/k-1]阶 Padé 逼近的零点的算子形式拓广为迭代函数的那一族迭代(k=1,2,…),还给出了误差的准确估计.     

在点估计判据下Euler级数、Euler迭代族以及Hauey迭代族的收敛性

论文证明了,当 S.Smale[1—3]的点估计判据α(f,z)=‖Df(z)~-1f(z)‖·(?)‖Df(z)~(-1)D~nf(z)/n!‖~(1/(n-1))≤3-22~(1/2)时,求 Banach 空间解析映照f零点ζ的 Newton 迭代的两族高阶推广以及ζ的逆级数都收敛,并且对其中每一个极限来说,条件中的常数3-22~(1/2)都是最好可能的.对其中以f在z的[1/k-1]阶 Padé 逼近的零点的算子形式拓广为迭代函数的那一族迭代(k=1,2,…),还给出了误差的准确估计.     

关于Riesz算子

引言.Dieudonne在[1](p.323问题5))中给出了复Banach空间Riesz算子的定义及充要条件。这个定义比笼统地称使Fredholm定理成立的算子为Riesz算子要具体,因而,也更有趣。显然这个定义是Dieudonne在深入地整理了Riesz紧算子理论之后而得到的。本文分两节:§1叙述Riesz算子的定义,比[1]约略简化,证明了二者同值;§2根据Riesz算子的定义导出了它的一些性质,说明对于Riesz算子,Fredholm定理依然成立。并且作为特例,包括了[2](p.430末两行)的命题。     

利用调和积分核求解二维狄氏问题

本文给出调和积分核的定义,证明它与狄氏问题的几个关系,然後应用於解决具体问题. §1.有调和函数U(r,θ),它的值在半径为R的圆周上已给定为f(θ),在圆内部之值U(r,θ)由柏桑公式给出     

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results. Received November 29, 1996 / Revised version received April 25, 1997     

Forcing sequences and inexact Newton iterates in Banach space

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately and in some unspecified manner. In the elegant paper [1], natural assumptions under which the forcing sequence is uniformly less than one were given based on the first Fréchet-derivative of the operator involved. Here, we use assumption on the second Fréchet-derivative. This way, we essentially reproduce all results found earlier. However, our upper error bounds on the distances involved are smaller.     

Adaptive Strategy for the Damping Parameters in an Iteratively Regularized Gauss–Newton Method

In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.     

Order Interval Secant Method for Nonlinear Systems

An order interval secant method is given. Its rate of convergence is faster than that of order interval Newton method in [1]. The existence and uniqueness of a solution to nonlinear systems and convergence of the interval iterative sequence are also proved.     

Projektive Newton-Verfahren und Anwendungen auf nichtlineare Randwertaufgaben

Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods.

     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

Local path-following property of inexact interior methods in nonlinear programming

We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of the original system. In case of an exact solution of the Newton system, it has been shown that the sequence of iterates is asymptotically tangent to the central path (Armand and Benoist in Math. Program. 115:199?C222, 2008). The purpose of the present paper is to extend this result to an inexact solution of the Newton system. We give quite general conditions on the different parameters of the algorithm, so that this asymptotic property is satisfied. Some numerical tests are reported to illustrate our theoretical results.     

序Banach空间中算子方程的迭代求解及其应用

利用半序理论和混合单调算子技巧,研究序Ｂａｎａｃｈ空间中非线性算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计．作为应用,讨论了序Ｂａｎａｃｈ空间中一类非线性积分方程的可解性,改进和推广了某些已知结果．     

Self-adaptive operator splitting methods for monotone variational inequalities

Summary. Solving a variational inequality problem VI(Ω,F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω,F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.     

集值映射的不动点指数与带间断非线性项的椭圆型方程的多重解

<正> 本文研究二阶半线性椭圆边值问题■的多重解(符号详见§3),其中φ(x,t)允许对t是不连续的.一些自由边界问题可以化归这类问题.为了统一处理φ(x,t)对t连续与不连续两种情形,我们采用集值映射的观点.为此推广了经典的算子与Hammerstein算子到集值映射,并发展了集值映射的Leray-Schauder度理论;与已有的集值映射理论不同,现在处理的是映射串(定