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1.
Tikhonov正则化方法是研究不适定问题最重要的正则化方法之一,但由于这种方法的饱和效应出现的太早,使得无法随着对解的光滑性假设的提高而提高正则逼近解的收敛率,也即对高的光滑性假设,正则解与准确解的误差估计不可能达到阶数最优.Schrroter T 和Tautenhahn U给出了一类广义Tikhonov正则化方法并重点讨论了它的最优误差估计, 但却未能对该方法的饱和效应进行研究.本文对此进行了仔细分析,并发现此方法可以防止饱和效应,而且数值试验结果表明此方法计算效果良好.  相似文献   

2.
金其年 《中国科学A辑》1999,29(8):715-723
对解非线性不适定问题的Tikhonov正则化证明了饱和性与一些逆结果 ,并考虑了正则化参数的最优后验选取 .  相似文献   

3.
对于带有右端扰动数据的第一类紧算子方程的病态问题 ,本文应用正则化子建立了一类新的正则化求解方法 ,称之为改进的Tikonov正则化 ;通过适当选取正则参数 ,证明了正则解具有最优的渐近收敛阶 .与通常的Tikhonov正则化相比 ,这种改进的正则化可使正则解取到足够高的最优渐近阶  相似文献   

4.
为获得病态线性方程组的高精度解,建立了一种优化模型,其最优解等价于早先提出的误差转移法和增广方程组法;指出后两者的本质机理是通过极小化解的模来近似极小化解的误差.为使算法适用于数据有污染的情况,进行了正则化改造.证明了新算法理论上与Tikhonov正则化等价.但当正则化参数趋于0时,目标函数的不同使得两者性能迥异,新算法可直接用于数据无污染的情况,而后者仍需选取合适的正则参数.数值算例验证了算法的有效性.  相似文献   

5.
王家军  李功胜 《数学季刊》2000,15(2):98-101
对于带有右扰动数据的第一类紧算子方程的病态问题。本文应用正则化子建立了一类新的正则化求解方法,称之为改进的Tikonov正则化;通过适当选取2正则参数,证明了正则解具有最优的渐近收敛阶,与通常的Tikhonov正则化相比,这种改进的正则化可使正则解取到足够高的最优渐近阶。  相似文献   

6.
基于混沌粒子群算法的Tikhonov正则化参数选取   总被引:2,自引:0,他引:2  
余瑞艳 《数学研究》2011,44(1):101-106
Tikhonov正则化方法是求解不适定问题最为有效的方法之一,而正则化参数的最优选取是其关键.本文将混沌粒子群优化算法与Tikhonov正则化方法相结合,基于Morozov偏差原理设计粒子群的适应度函数,利用混沌粒子群优化算法的优点,为正则化参数的选取提供了一条有效的途径.数值实验结果表明,本文方法能有效地处理不适定问题,是一种实用有效的方法.  相似文献   

7.
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   

8.
陈宏  侯宗义 《中国科学A辑》1994,37(8):808-814
讨论了解算子与右端都近似给定的第一类算子方程的迭代Tikhonov正则化方法,建立了一种选择正则参数的方法——广义Arcangeli方法,得到正则化逼近解的收敛速度估计。  相似文献   

9.
在Tikhonov正则化方法的基础上将其转化为一类l1极小化问题进行求解,并基于Bregman迭代正则化构建了Bregman迭代算法,实现了l1极小化问题的快速求解.数值实验结果表明,Bregman迭代算法在快速求解算子方程的同时,有着比最小二乘法和Tikhonov正则化方法更高的求解精度.  相似文献   

10.
非线性不适定问题的Tikhonov正则化的参数选取方法   总被引:1,自引:0,他引:1  
在Tikhonov正则化中,如何选取正则参数极为重要,直至现在,仍有许多问题期待解决.本文对非线性不适定问题考虑了Tikhonov正则化,提出了一个新的简单的正则参数的最优选取法,并对由此得到的正则参数,研究了Tikhonov正则化解的收敛性,并且当x-最小范数解满足“源条件”时,在适当的条件下,导出了最优收敛率.  相似文献   

11.
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.

  相似文献   


12.
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   

13.
In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.  相似文献   

14.
《Applied Mathematical Modelling》2014,38(19-20):4686-4693
In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.  相似文献   

15.
In this study we prove a stability estimate for an inverse heat source problem in the n-dimensional case. We present a revised generalized Tikhonov regularization and obtain an error estimate. Numerical experiments for the one-dimensional and two-dimensional cases show that the revised generalized Tikhonov regularization works well.  相似文献   

16.
The computation of an approximate solution of linear discrete ill-posed problems with contaminated data is delicate due to the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution to errors in the data by replacing the given ill-posed problem by a nearby problem, whose solution is less sensitive to perturbation. This regularization method requires that a suitable value of the regularization parameter be chosen. Recently, Brezinski et al. (Numer Algorithms 49, 2008) described new approaches to estimate the error in approximate solutions of linear systems of equations and applied these estimates to determine a suitable value of the regularization parameter in Tikhonov regularization when the approximate solution is computed with the aid of the singular value decomposition. This paper discusses applications of these and related error estimates to the solution of large-scale ill-posed problems when approximate solutions are computed by Tikhonov regularization based on partial Lanczos bidiagonalization of the matrix. The connection between partial Lanczos bidiagonalization and Gauss quadrature is utilized to determine inexpensive bounds for a family of error estimates. In memory of Gene H. Golub. This work was supported by MIUR under the PRIN grant no. 2006017542-003 and by the University of Cagliari.  相似文献   

17.
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   

18.
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   

19.
In this paper we consider nonlinear ill-posed problems F(x) = y 0, where x and y 0 are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.  相似文献   

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