基于埃特金加速的改进Landweber迭代正则化方法

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.     

图像恢复的一种快速迭代正则化方法

本文研究了将图像恢复问题转化为大型的线性不适定问题的求解.利用由Landweber迭代正则化方法改进所得到的快速收敛的迭代正则化方法,处理具有可分离点扩散函数的图像恢复问题.图像恢复实验表明该方法可大大提高收敛速度,且在计算中只需要较少的存储量.     

约束Fractional Tikhonov正则化的模迭代方法

反问题是现在数学物理研究中的一个热点问题,而反问题求解面临的一个本质性困难是不适定性。求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法.如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.当前,最为流行的正则化方法有基于变分原理的Tikhonov正则化及其改进方法,此类方法是求解不适定问题的较为有效的方法,在各类反问题的研究中被广泛采用,并得到深入研究.     

一类逆时反问题的改进正则化方法的收敛性

1引 言 非线性反问题广泛地存在于许多科学和工程问题中,反问题求解的主要困难在于问题的不适定性,即待求函数或参量不连续依赖于观测数据.用来求解非线性不适定问题的方法主要有Tikhonov正则化方法和迭代正则化方法[1,2,3,4].Tikhonov正则化方法是通过引入正则化参数及稳定泛函,将目标泛函离散化,从而得到解的一个稳定近似,即正则化解.     

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.     

基于混沌粒子群算法的Tikhonov正则化参数选取

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.     

解第一类算子方程的一种迭代正则化方法

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.     

一类球型区域上变系数反向热传导问题

考虑了一类球型区域上变系数反向热传导问题.这个问题是不适定的,即问题的解（若存在）并不连续依赖于测量数据.构造了投影迭代正则化方法,得到了该反问题的正则近似解,同时给出了在先验和后验参数选取规则下精确解与正则近似解之间的收敛性误差估计.最后,通过数值结果验证了该方法的有效性.     

求解一类Helmholtz方程Cauchy问题的中心差分正则化方法

Helmholtz方程Cauchy问题是严重不适定问题,本文我们在一个带形区域上考虑了一类Helmholtz方程Cauchy问题:已知Cauchy数据u(0,y)=g(y),在区间0＜x＜1上求解.我们用半离散的中心差分方法得到了这一问题的正则化解,给出了正则化参数的选取规则,得到了误差估计.     

基于Tikhonov正则化的扩展Kaczmarz算法

为了更加有效地处理不适定问题,在扩展Kaczmarz算法的思想基础上,提出一种基于Tikhonov正则化的最大残差控制的扩展Kaczmarz算法并证明其收敛性.利用sheep-logan头部图像等进行图像重建实验.数値结果表明,该算法和最大残差控制的扩展Kaczmarz算法(MREK算法)相比,误差更小,图像质量更优.     

A Modified Tikhonov Regularization for Linear Operator Equations

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator. With some accurate estimates in the solution errors, optimal convergence order of the regularized solutions is obtained by a priori choice of the regularization parameter. Furthermore, numerical results are given for several ill-posed integral equations, which not only roughly coincide with the theoretical results but also show that MTR can be more accurate than ordinary Tikhonov regularization (OTR).     

Some results on the regularization of LSQR for large-scale discrete ill-posed problems

LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems (CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition (TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory predicts, but they are not for mildly ill-posed problems and additional regularization is needed.     

General Regularizing Functionals for Solving Ill-Posed Problems in Lebesgue Spaces

We study sufficient conditions for general integral functionals in Lebesgue spaces to possess regularizing properties required for solving nonlinear ill-posed problems. We select special classes of such functionals: uniformly convex and quasiuniformly convex (in the extended sense). We give a series of examples and, in particular, a functional that can be used in a generalized version of the maximum entropy method in Lebesgue spaces.     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

On the order optimality of the regularization via inexact Newton iterations

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.     

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).     

On application of asymptotic generalized discrepancy principle to the analysis of epidemiology models

Various implementations of the discrepancy principle (DP) for linear ill-posed problems are given in a large number of papers. In all of these papers, the DP has been justified for special types of regularization strategies. In our paper, a unified approach to the construction of the DP is presented that does not require any special structure of the regularizing operator. In that respect, the new method generalizes all prior results on the DP principle for linear irregular operator equations with noisy data. The efficiency of the proposed scheme is demonstrated for a parameter identification problem in avian influenza. In solving this particular inverse problem, it turned out to be beneficial to use some regularization strategies, for which the earlier (structure-based) discrepancy principles were not applicable. This motivated the development of a novel DP put forth in the current paper.     

A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.