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.     

On Lanczos Based Methods for the Regularization of Discrete Ill-Posed Problems

We study hybrid methods for the solution of linear ill-posed problems. Hybrid methods are based on he Lanczos process, which yields a sequence of small bidiagonal systems approximating the original ill-posed problem. In a second step, some additional regularization, typically the truncated SVD, is used to stabilize the iteration. We investigate two different hybrid methods and interpret these schemes as well-known projection methods, namely least-squares projection and the dual least-squares method. Numerical results are provided to illustrate the potential of these methods. This gives interesting insight in to the behavior of hybrid methods in practice.This revised version was published online in October 2005 with corrections to the Cover Date.     

Steepest Descent, CG, and Iterative Regularization of Ill-Posed Problems

The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

Examples of Saturated Convergence Rates for Tikhonov Regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data (both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u ₊ (i.e.the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u ₊ im(A*A). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with only for some values of , in general not greater than one. This phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.This revised version was published online in October 2005 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.     

Convergence and Optimality of Adaptive Regularization for Ill-posed Deconvolution Problems in Infinite Spaces

The adaptive regularization method is first proposed by Ryzhikov et al. in [6] for the deconvolution in elimination of multiples which appear frequently in geoscience and remote sensing. They have done experiments to show that this method is very effective. This method is better than the Tikhonov regularization in the sense that it is adaptive, i.e., it automatically eliminates the small eigenvalues of the operator when the operator is near singular. In this paper, we give theoretical analysis about the adaptive regularization. We introduce an a priori strategy and an a posteriori strategy for choosing the regularization parameter, and prove regularities of the adaptive regularization for both strategies. For the former, we show that the order of the convergence rate can approach O（｜｜n｜｜^4v/4v＋1） for some 0 〈 v 〈 1, while for the latter, the order of the convergence rate can be at most O（｜｜n｜｜^2v/2v＋1） for some 0 〈 v 〈 1.     

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.     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

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.     

Well-Posedness by Perturbations of Variational Problems

In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.