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1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
The Landweber scheme is a method for algebraic image reconstructions. The convergence behavior of the Landweber scheme is of both theoretical and practical importance. Using the diagonalization of matrix, we derive a neat iterative representation formula for the Landweber schemes and consequently establish the convergence conditions of Landweber iteration. This work refines our previous convergence results on the Landweber scheme. 相似文献
9.
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
10.
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
11.
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
12.
Andreas Neubauer 《Numerical Functional Analysis & Optimization》2018,39(6):737-762
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. 相似文献
13.
Recovering an unknown low-rank or approximately low-rank matrix from a sampling set of its entries is known as the matrix completion problem. In this paper, a nonlinear constrained quadratic program problem concerning the matrix completion is obtained. A new algorithm named the projected Landweber iteration (PLW) is proposed, and the convergence is proved strictly. Numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of the unknown low-rank matrix. 相似文献
14.
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
15.
In this paper we propose the R-K type Landweber iteration and investigate the convergence of the method for nonlinear ill-posed problem F(x)=y where F:H→H is a nonlinear operator between Hilbert space H . Moreover, for perturbed data with noise level δ we prove that the convergence rate is O(δ2/3) under appropriate conditions. Finally, the numerical performance of this R-K type Landweber iteration for a nonlinear convolution equation is compared with the Landweber iteration. 相似文献
16.
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
17.
Martin Hanke 《Numerical Functional Analysis & Optimization》2013,34(11):1500-1510
We reconsider the Landweber iteration for nonlinear ill-posed problems. It is known that this method becomes a regularization method in the case when the iteration is terminated as soon as the residual drops below a certain multiple of the noise level in the data. So far, all known estimates of this factor are greater than two. Here we derive a smaller factor that may be arbitrarily close to one depending on the type of nonlinearity of the underlying operator equation. 相似文献
18.
Linjun Wang Xu HanJiujiu Chen 《Journal of Computational and Applied Mathematics》2011,235(14):4083-4094
We present an improved iteration regularization method for solving linear inverse problems. The algorithm considered here is detailedly given and proved that the computational costs for the proposed method are nearly the same as the Landweber iteration method, yet the number of iteration steps by the present method is even less. Meanwhile, we obtain the optimum asymptotic convergence order of the regularized solution by choosing a posterior regularization parameter based on Morozov’s discrepancy principle, and the present method is applied to the identification of the multi-source dynamic loads on a surface of the plate. Numerical simulations of two examples demonstrate the effectiveness and robustness of the present method. 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