非线性不适定问题一种双循环的牛顿型迭代格式

本文研究非线性算子方程F(x)=y的解,结合最速下降法,Newton-Landweber迭代格式及正则化思想,在F满足适当的条件下,构造出新的双循环迭代格式.本文对格式的收敛性进行了严格论证,并估计出迭代格式的收敛精度.     

中点牛顿迭代格式的最优性

研究牛顿迭代法的变形格式,在中点迭代格式的基础上,提出了如下形式的一般迭代格式:{P∶zk+1=(xk-f(xk))/(f′(xk)) C∶xk+1=xk-(f(xk))/(f′(μxk+(1-μ)zk+1))并证明了中点迭代格式是这类迭代格式中最优的,收敛阶为3.     

守恒型扩散方程非线性离散格式的性质分析和快速求解

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.     

Banach空间中半线性发展方程的复合单调迭代技巧

本文对具有正则锥的序Banach空间中半线性发展方程，利用C0-半群理论和复合单调迭代技巧，构造出了抽象迭代格式，并建立了最大、最小(拟)解的存在性．     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).     

求解非线性函数零点的一种迭代格式

本文就求解相当广泛的一类函数的零点问题，建立了一种简单而有效的迭代格式，证明了迭代格式的收敛性并给出了几个实际例子．     

Hilbert空间中非扩张余弦族的显式、隐式和黏性迭代

研究了Hilbert空间中一些逼近单参数非扩张余弦族公共不动点的迭代格式.借助余弦族理论,在较弱的条件下分别对显式、隐式和黏性的迭代过程建立了一系列的收敛定理.结果表明上述三种迭代过程适用于非扩张余弦族;并且隐式和黏性迭代格式在收敛性上优越于显式迭代格式.     

关于实李普希兹映射的lshikawa迭代的收敛率和加速收敛问题

本文研究使用Ｉｓｈｉｋａｗａ迭代格式求实李普希兹映射的不动点，指出该迭代格式仅具线性收敛率；对于参变量序列｛αｎ｝、｛βｎ｝所取的不同的值，比较了迭代格式的收敛速度．在给出加速因子定义的基础上，本文给出了加速收敛的一个充分条件．     

关于非线性方程的牛顿迭代格式初始值选取的注记

基于构造非线性方程的牛顿迭代格式简便和牛顿迭代格式具有收敛快的特点,在解决实际问题时,牛顿迭代格式显得尤为重要,但是,牛顿迭代格式的初始值选取具有很大的局限性.利用泰勒级数展开,对牛顿迭代格式的收敛性进行分析,从而提出改进牛顿迭代格式的初始值选取方案,并利用不同的数值算例验证牛顿迭代格式收敛区域的改进方案的可行性,同时数值算例表明该方法具有操作简单的特点.     

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.     

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     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.     

Modified Landweber Iteration in Banach Spaces—Convergence and Convergence Rates

We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example.     

Multilevel augmentation methods for solving the sine-Gordon equation

In this paper we develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

On convergence rates for the Iteratively regularized Gauss-Newton method

Received on 3 February 1995. Revised on 20 April 1996. In this paper we prove that the iteratively regularized Gauss-Newtonmethod is a locally convergent method for solving nonlinearill-posed problems, provided the nonlinear operator satisfiesa certain smoothness condition. For perturbed data we proposea priori and a posteriori stopping niles that guarantee convergenceof the iterates, if the noise level goes to zero. Under appropriatecloseness and smoothness conditions on the exact solution weobtain the same convergence rates as for linear ill-posed problems.     

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.     

非线性不适定算子方程的双参数正则化方法

对非线性不适定算子方程,引入一种双参数正则化方法求解,讨论了这种正则化方法解的存在性、稳定性和收敛性.     

Hermite subdivision on manifolds via parallel transport

We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C ¹ smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from.     

A New Gradient Method for Ill-Posed Problems

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.     

非线性不适定算子方程算子与右端项皆有扰动的Landweber迭代法

考虑非线性算子方程 F（x）=y（1）其中 F:D（F）　 X→Y,X,Y为 Hilbert空间.F是 Frechet可微的。 这里考虑算子方程的解x+不连续依赖于右端数据的情况。由于不稳定性,并且在实际问题中只有近似数据yδ满足 ‖yδ-y‖≤δ（2）可以得到,方程（1）必须正则化.