求解病态问题的一种改进的Tikhonov正则化：⑴正则化方法的建立

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。     

改进的Tikhonov 正则化及其正则解的最优渐近阶估计

对于算子与右端都有扰动的第一类算子方程建立了一类新的正则化方法(称为改进的 Ｔｉｋｈｏｎｏｖ正则化)．应用紧算子的奇异系统和广义 Ａｒｃａｎｇｅｌｉ方法后验选取正则参数,证明了正则解具有最优的渐近阶并给出了相应的算例分析．     

关于迭代Tikhonov正则化的最优正则参数选取

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。     

一种新的正则化方法的正则参数的最优后验选取

应用紧算子的奇异系统和广义Arcangeli方法后验选取正则参数，证明了文[1]中所建立的求解第一类算子方程的正则化方法是收敛的，且正则解具有最优的渐近阶。     

解第一类算子方程的一种新的正则化方法

对算子与右端都为近似给定的第一类算子方程提出一种新的正则化方法，依据广义Ａｒｃａｎｇｅｌｉ方法选取正则参数，建立了正则解的收敛性。这种新的正则化方法与通常的Ｔｉｋｈｏｎｏｖ正则化方法相比较，提高了正则解的渐近阶估计。     

应用正则化子建立求解不适定问题的正则化方法的探讨

根据紧算子的奇异系统理论,提出一种新的正则化子进而建立了一类新的求解不适定问题的正则化方法。分别通过正则参数的先验选取和后验确定方法,证明了正则解的收敛性并得到了其最优的渐近收敛阶;验证了应用Newton迭代法计算最佳参数的可行性。最后建立了当算子与右端均有扰动时相应的正则化求解策略。文中所述方法完善了一般优化正则化策略的构造理论。     

不适定问题的迭代Tikhonov正则化方法

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,使得不可能随着解的光滑性假设的提高而提高收敛率,即不能使正则解与准确解的误差估计达到阶数最优．本文所讨论的迭代的Tikhonov正则化方法对此进行了改进,保证了误差估计总可以达到阶数最优．数值试验结果表明计算效果良好．     

基于奇异值分解建立的一种新的正则化方法

根据紧算子的奇异系统理论，引入一种正则化滤子函数，从而建立一种新的正则化方法来求解右端近似给定的第一类算子方程，并给出了正则解的误差分析。通过正则参数的先验选取，证明了正则解的误差具有渐进最优阶。      

非线性不适定问题的正则化方法

本文考虑非线性不适定问题Tx=y的近似求解,利用Тихоноь正则化方法来逼近问题的x-极小模解,当算子和右端都近似已知时,给出一种决定正则化参数的方法,并给出正则解的收效性和渐近收敛阶估计。     

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.     

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.     

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).     

Regularization of differential-algebraic equations

Linear systems of ordinary differential equations with an identically singular or rectangular matrix multiplying the derivative of the unknown vector function are numerically solved by applying the least squares method and Tikhonov regularization. The deviation of the solution of the regularized problem from the solution set of the original problem is estimated depending on the regularization parameter.     

Regularization of nonlinear illposed problems with closed operators

In this paper Tikhonov regularization for nonlinear illposed problems is investigated. The regularization term is characterized by a closed linear operator, permitting seminorm regularization in applications. Results for existence, stability, convergence and con- vergence rates of the solution of the regularized problem in terms of the noise level are given. An illustrating example involving parameter estimation for a one dimensional stationary heat equation is given.     

Regularization of Discrete Ill-Posed Problems

The paper concerns conditioning aspects of finite-dimensional problems arising when the Tikhonov regularization is applied to discrete ill-posed problems. A relation between the regularization parameter and the sensitivity of the regularized solution is investigated. The main conclusion is that the condition number can be decreased only to the square root of that for the nonregularized problem. The convergence of solutions of regularized discrete problems to the exact generalized solution is analyzed just in the case when the regularization corresponds to the minimal condition number. The convergence theorem is proved under the assumption of the suitable relation between the discretization level and the data error. As an example the method of truncated singular value decomposition with regularization is considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tikhonov Regularization of Large Linear Problems

Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a new numerical method, based on Lanczos bidiagonalization and Gauss quadrature, for Tikhonov regularization of large-scale problems. An estimate of the norm of the error in the data is assumed to be available. This allows the value of the regularization parameter to be determined by the discrepancy principle.     

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

Computation of control for a controlled partial differential equation is a di?cult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.     

Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.     

Regularization Using QR Factorization and the Estimation of the Optimal Parameter

In this paper we propose a direct regularization method using QR factorization for solving linear discrete ill-posed problems. The decomposition of the coefficient matrix requires less computational cost than the singular value decomposition which is usually used for Tikhonov regularization. This method requires a parameter which is similar to the regularization parameter of Tikhonov's method. In order to estimate the optimal parameter, we apply three well-known parameter choice methods for Tikhonov regularization.This revised version was published online in October 2005 with corrections to the Cover Date.