求解不适定问题的快速Landweber迭代法

本文从一般迭代法的级数形式出发，将一般迭代法的每一步分解为矩阵计算和求解两步，并对其中的矩阵计算部分进行了修改，在此基础上提出了快速迭代法，最后通过数值实验验证了我们的算法不仅提高了计算速度，同时也大大减少了计算量，是一种效率很高的算法。     

解不适定算子方程的一个定常二步隐式迭代法

１．引言 设Ｘ,Ｙ是两个Ｈｉｌｂｅｒｔ空间,Ａ：Ｘ→Ｙ是有界线性算子,考虑算子方程 Ａｘ＝ｙ（１．１）如果Ａ的值域Ｒ（Ａ）在Ｙ中非闭,则方程（１．１）是不适定的［１］．许多应用科学中都归结出这一类方程,特别地,许多反问题是不适定的［２,３］．本文考虑方程（１．１）的 Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ广义解,这里Ａ是算子Ａ的Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ广义逆［１］．Ａ＋ｙ存在当且仅当本文均作这一假设．在实际中,通常代替（１．１）的是扰动方程这里右端项,为一给定的误差水平,Ｑ是Ｙ到Ｒ（Ａ）的正交投影算子．对扰…     

求解非线性不适定的混合Newton-Tikhonov迭代法

鉴于Newton型方法在实际计算中计算量可能非常大,因此提出了一种一步Newton结合若干步简化Newton的混合Newton-Tikhonov方法,并且在一定条件下证明了该方法的收敛性和稳定性.数值试验表明,在减少计算量方面该方法相对于经典的Newton方法有明显的改善.     

解不适定算子方程的一个定常二步隐式迭代法

On the basis of an implicit iterative method for ill-posed operator equations,we introduce a relaxation factor and a weighted factor , and obtain a stationarytwo-step implicit iterative method. The range of the factors which guarantee theconvergence of iteration is explored.We also study the convergence properties and ratesfor both non-perturbed andperturbed equations.An implementable algorithm is presented by using Morozov discrepancy principle.The theoretical results show that the convergence rates of the new methods always lead to optimal convergentrates which are superior to those of the original one after choosing suitable relaxation and weightedfactors. Numerical examplesare also given, which coincide well with the theoretical results.     

解线性不适定方程隐式迭代法的r-步拟残差准则

本文给出了解不适定算子方程隐式迭代法后验选取步数的一类准则,称为r-步拟残差准则,证明了它们总导致最佳收敛阶.这类准则包含著名的Morozov残差准则和Gfrerer准则作为特例.     

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.     

解线性不适定方程隐式迭代法的,γ-步拟残差准则

本文给出了解不适定算子方程隐式迭代法后验选取步数的一类准则,称为,ｒ－步拟残差准则,证明了它们总导致最佳收敛阶,这类准则包含著名的 Ｍｏｒｏｚｏｖ残差准则和 Ｇｆｒｅｒｅｒ 准则作为特例．     

基于正交多项式的解不适定算子方程的隐式迭代法

该文研究了基于Chebyshev和Jacobi多项式的解不适定算子方程的隐式迭代法.建立了隐式迭代法和由Hanke提出的显式迭代法之间的关系. 给出了与Chebyshev第一和第二多项式相关的迭代格式的残差有理式的一个重要引理. 对精确和扰动的数据, 研究了方程的收敛性和收敛速率. 利用Morozov残差原则, 给出了一个可执行的强健的正则化算法.最后还给出了一些数值例子, 数值结果与理论分析基本一致.     

Hilbert尺度下求解非线性不适定问题的多水平迭代法

本文吸取了多水平方法的思想，采用多水平方法提供了离散化参数和迭代初值的合理的选择方法，提出了Hilbert尺度下求解非线性不适定问题的多水平Landweber迭代算法，并给出了算法的收敛性分析，证明了算法在整体上提高了Hilbert尺度下的Landweber迭代法的迭代效率。     

解非线性不适定算子方程的一种正则化Newton迭代法

本文探讨一种求解非线性不适定算子方程的正则化Newton迭代法.本文讨论了这种迭代法在一般条件下的收敛性以及其他的一些性质.这种迭代法结合确定迭代次数的残差准则有局部收敛性.     

A Regularized Iterative Scheme for Solving Nonlinear Ill-Posed Problems

In this article, we consider a regularized iterative scheme for solving nonlinear ill-posed problems. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. The iterative scheme is stopped using an a posteriori stopping rule, and we prove that the scheme converges to the solution of the well-known Lavrentiev scheme. The salient features of the proposed scheme are: (i) convergence and error estimate analysis require only weaker assumptions compared to standard assumptions followed in literature, and (ii) consideration of an adaptive a posteriori stopping rule and a parameter choice strategy that gives the same convergence rate as that of an a priori method without using the smallness assumption, the source condition. The above features are very useful from theory and application points of view. We also supply the numerical results to illustrate that the method is adaptable. Further, we compare the numerical result of the proposed method with the standard approach to demonstrate that our scheme is stable and achieves good computational output.     

Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems

This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Hölder-type sourcewise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.     

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that the imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.     

The Optimal Approximations for Solving Linear Ill-Posed Problems

New projection discrete schemes for ill-posed problems are constructed. We show that for equations with self-adjoint operators the use of self-adjoint projection schemes is not optimal in the sense of the amount of discrete information.     

Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems

For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejér property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed.     

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweberiteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numericalperformance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared withthat of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based onthe same convergence accuracy.     

Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators

The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 271–276, February, 2005.     

非线性问题的插值摄动解法

本文用插值摄动法[1]求解几个非线性问题.算例表明,本文方法有很好的精度.