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1.
Frozen Landweber Iteration for Nonlinear Ill-Posed Problems 总被引:1,自引:0,他引:1
J. Xu B. Han L. Li 《应用数学学报(英文版)》2007,23(2):329-336
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. 相似文献
2.
In this paper, we investigate the convergence behavior of a Runge–Kutta type modified Landweber method for nonlinear ill-posed operator equations. In order to improve the stability and convergence of the Landweber iteration, a 2-stage Gauss-type Runge–Kutta method is applied to the continuous analogy of the modified Landweber method, to give a new modified Landweber method, called R–K type modified Landweber method. Under some appropriate conditions, we prove the convergence of the proposed method. We conclude with a numerical example confirming the theoretical results, including comparisons to the modified Landweber iteration. 相似文献
3.
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
4.
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
5.
In this paper, we are interested in the solution of nonlinear inverse problems of the form F(x)=y. We propose an implicit Landweber method, which is similar to the third-order midpoint Newton method in form, and consider the convergence behavior of the implicit Landweber method. Using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. We conclude with numerical examples confirming the theoretical results, including comparisons with the classical Landweber iteration and presented modified Landweber methods. 相似文献
6.
Qinian Jin 《Numerische Mathematik》2012,121(2):237-260
Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing increments by regularizing local linearized equations. The method is terminated by a discrepancy principle. In this paper we consider the inexact Newton regularization methods with the inner scheme defined by Landweber iteration, the implicit iteration, the asymptotic regularization and Tikhonov regularization. Under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of Rieder. We in fact obtain a more general order optimality result by considering these inexact Newton methods in Hilbert scales. 相似文献
7.
In this paper, we study the convergence and the convergence rates of an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. Opposed to the traditional methods, we analyze an inexact Newton–Landweber iteration depending on the Hölder continuity of the inverse mapping when the data are not contaminated by noise. With the namely Hölder-type stability and the Lipschitz continuity of DF, we prove convergence and monotonicity of the residuals defined by the sequence induced by the iteration. Finally, we discuss the convergence rates. 相似文献
8.
Otmar Scherzer 《Numerische Mathematik》1998,80(4):579-600
Summary. The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently
by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates
developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations,
when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper
is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented
until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization.
An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The
numerical performance of this multi level algorithm is compared with Landweber's iteration.
Received October 21, 1996 / Revised version received July 28, 1997 相似文献
9.
Tomas Johansson Daniel Lesnic 《Numerical Methods for Partial Differential Equations》2007,23(5):998-1017
In this article, an iterative algorithm based on the Landweber‐Fridman method in combination with the boundary element method is developed for solving a Cauchy problem in linear hydrostatics Stokes flow of a slow viscous fluid. This is an iteration scheme where mixed well‐posed problems for the stationary generalized Stokes system and its adjoint are solved in an alternating way. A convergence proof of this procedure is included and an efficient stopping criterion is employed. The numerical results confirm that the iterative method produces a convergent and stable numerical solution. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 相似文献
10.
Archiv der Mathematik - The convergence analysis of the Landweber iteration for solving inverse problems in Banach spaces via Hölder stability estimates is well studied by de Hoop et al.... 相似文献
11.
Andreas Neubauer 《Numerische Mathematik》2000,85(2):309-328
Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index
for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems
in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces
by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions
needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented
confirming the theoretical ones.
Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999 相似文献
12.
13.
This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regularization can quicken the convergence speed and reduce the calculation burden efficiently. 相似文献
14.
O. Scherzer 《Applied Mathematics and Optimization》1998,38(1):45-68
In this paper a convergence analysis for a modified Landweber iteration for the solution of nonlinear ill-posed problems
is presented. A priori and a posteriori stopping criteria for terminating the iteration are compared. Some numerical results
for the solution of a parameter estimation problem are presented.
Accepted 11 September 1996 相似文献
15.
Torsten Hein 《Numerical Functional Analysis & Optimization》2013,34(10):1158-1184
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. 相似文献
16.
Summary.
In this paper we prove that the Landweber iteration is a stable
method for solving nonlinear ill-posed problems. For perturbed data with
noise level we propose a stopping rule that yields the
convergence rate
)
under appropriate conditions. We
illustrate these conditions for a few examples.
Received
February 15, 1993 / Revised version received August 2, 1994 相似文献
17.
《Mathematical and Computer Modelling》2006,43(7-8):892-909
This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals. 相似文献
18.
This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017 相似文献
19.
We consider a linear steady-state eddy-current problem for a magnetic field in a bounded domain. The boundary consists of two parts: reachable with prescribed Cauchy data and unreachable with no data on it. We design an iterative (Landweber type) algorithm for solution of this problem. At each iteration step two auxiliary mixed well-posed boundary value problems are solved. The analysis of temporary problems is performed in suitable function spaces. This creates the basis for the convergence argument. The theoretical results are supported with numerical experiments. 相似文献
20.
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。 相似文献