共查询到20条相似文献,搜索用时 15 毫秒
1.
H. Chen & Z. Y. Hou 《计算数学(英文版)》1996,14(2):108-119
1.IntroductionLetX,YberealHilbertspaces,T:X-Yab0ulldedlinear0perat0rwithnonclosedrangeR(T),yED(T+)=R(T)+R(T)",whereT+istheM0ore-PenroseinverseofTl11.Foreachb>0,lety6EYbesuchthatIly-ueIls6.(1)Asweknow,theproblemofsolvingtheoperatorequationofthefirstkindTx=y(2)..is,ingenerality,ill-p0sedl2].Alsowecann0tensurethatT+Wisareas0nableapproximation0fT+ysinceT+isaUnbounded0perat0r-Inpractice,onetriestoc0nstructastableaPprokimate6olutiontotheequation(2)byregularizationmeth0ds.Awell-knownregu… 相似文献
2.
提出了一种求解第一类算子方程的新的迭代正则化方法,并依据广义Arcangeli方法选取正则参数,建立了正则解的收敛性.与通常的Tikhonov正则化方法相比较,提高了正则解的渐近阶估计. 相似文献
3.
考虑了第一类Fredholm积分方程的求解.采用有矩阵压缩策略的多尺度配置方法来离散Lavrentiev迭代方程,在积分算子是弱扇形紧算子时,给出近似解的先验误差估计,并给出了改进的后验参数的选择方法,得到了近似解的收敛率.最后,举例说明算法的有效性. 相似文献
4.
关于迭代Tikhonov正则化的最优正则参数选取 总被引:2,自引:0,他引:2
本文讨论了算子和右端都近似给定的第一类算子方程的迭代Tikhonov正则化,给出了不依赖于准确解的任何信息但能得到最优收敛阶的正则参数选取法。 相似文献
5.
6.
We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear
ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an
inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and
unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of
schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients. 相似文献
7.
提出了一种新的解第一类算子方程的迭代正则化方法,与通常的迭代正则化方法相比,提高了j次迭代正则解的渐近阶估计.同时,给出了后验正则化参数的选择. 相似文献
8.
Marco Donatelli 《Numerical Algorithms》2012,60(4):651-668
Nonstationary iterated Tikhonov is an iterative regularization method that requires a strategy for defining the Tikhonov regularization parameter at each iteration and an early termination of the iterative process. A classical choice for the regularization parameters is a decreasing geometric sequence which leads to a linear convergence rate. The early iterations compute quickly a good approximation of the true solution, but the main drawback of this choice is a rapid growth of the error for later iterations. This implies that a stopping criteria, e.g. the discrepancy principle, could fail in computing a good approximation. In this paper we show by a filter factor analysis that a nondecreasing sequence of regularization parameters can provide a rapid and stable convergence. Hence, a reliable stopping criteria is no longer necessary. A geometric nondecreasing sequence of the Tikhonov regularization parameters into a fixed interval is proposed and numerically validated for deblurring problems. 相似文献
9.
In this paper, we use the idea of Kantorovich regularization to develop the fast multiscale Kantorovich method and the fast iterated multiscale Kantorovich method. For some kinds of weakly singular integral equations with nonsmooth inhomogeneous terms, we show that our two proposed methods can still obtain the optimal order of convergence and superconvergence order, respectively. Numerical examples are given to demonstrate the efficiency of the methods. 相似文献
10.
In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.
11.
Alessandro Buccini Marco Donatelli Lothar Reichel 《Numerical Linear Algebra with Applications》2017,24(4)
Tikhonov regularization is one of the most popular approaches to solving linear discrete ill‐posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method. 相似文献
12.
We consider the nonstationary iterated Tikhonov regularization in Banach spaces which defines the iterates via minimization problems with uniformly convex penalty term. The penalty term is allowed to be non-smooth to include \(L^1\) and total variation (TV) like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and discontinuities in practical applications. We present the detailed convergence analysis and obtain the regularization property when the method is terminated by the discrepancy principle. In particular we establish the strong convergence and the convergence in Bregman distance which sharply contrast with the known results that only provide weak convergence for a subsequence of the iterative solutions. Some numerical experiments on linear integral equations of first kind and parameter identification in differential equations are reported. 相似文献
13.
Tikhonov regularization is a popular method for the solution of linear discrete ill-posed problems with error-contaminated data. Nonstationary iterated Tikhonov regularization is known to be able to determine approximate solutions of higher quality than standard Tikhonov regularization. We investigate the choice of solution subspace in iterative methods for nonstationary iterated Tikhonov regularization of large-scale problems. Generalized Krylov subspaces are compared with Krylov subspaces that are generated by Golub–Kahan bidiagonalization and the Arnoldi process. Numerical examples illustrate the effectiveness of the methods. 相似文献
14.
Nonstationary Iterated Tikhonov Regularization 总被引:5,自引:0,他引:5
A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. It is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number. 相似文献
15.
For ordinary and iterated Tikhonov regularization of linear ill-posed problems, we propose a parameter choice strategy that leads to optimal (super-) convergence rates for certain linear functionals of the regularized solution. It is not necessary to know the smoothness index of the exact solution; approximate knowledge of the smoothness index for the linear functional suffices 相似文献
16.
MA Qinghua & WANG Yanfei Department of Information Sciences College of Arts Science of Beijing Union University Beijing China National Key Laboratory on Remote Sensing Science Institute of Remote Sensing Applications Chinese Academy of Sciences Beijing China Department of Mathematics University of Central Florida P. O. Box Orlando FL - USA 《中国科学A辑(英文版)》2005,48(4):485-492
The adaptive regularization method is first proposed by Ryzhikov et al. for the deconvolution in elimination of multiples. This method is stronger than the Tikhonov regularization in the sense that it is adaptive, i.e. it eliminates the small eigenvalues of the adjoint operator when it is nearly singular. We will show in this paper that the adaptive regularization can be implemented iterately. Some properties of the proposed non-stationary iterated adaptive regularization method are analyzed. The rate of convergence for inexact data is proved. Therefore the iterative implementation of the adaptive regularization can yield optimality. 相似文献
17.
AbstractIn this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L2-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L2-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results. 相似文献
18.
余瑞艳 《数学的实践与认识》2014,(10)
为克服Landweber迭代正则化方法在求解大规模不适定问题时收敛速度慢的不足,将埃特金加速技巧与不动点迭代相结合,构建了能快速收敛的改进Landweber迭代正则化方法.数值实验结果表明:改进的迭代正则化方法在稳定求解不适定问题时,能够快速地收敛至问题的最优解,较Landweber迭代正则化方法大大提高了收敛速度. 相似文献
19.
Robert Plato 《Numerische Mathematik》1996,75(1):99-120
Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative
methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition
with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle
(being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence
rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive
semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition
contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev.
A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented.
Received August 29, 1994 / Revised version received September 19, 1995 相似文献
20.
Heinz W. Engl 《Numerical Functional Analysis & Optimization》2013,34(2):201-222
In this paper, we give necessary and sufficient conditions for weak and strong convergence of general regularization methods for the solution of linear ill-posed problems in Hilbert space. As special cases we obtain convergence criteria for Tychonoff regularization, for a regularization method based on Showalter's integral formula, and for regularization by truncated iteration 相似文献