Tikhonov Regularization Of Ill-posed Problems Research Articles

On the existence and stability of minimizers for generalized Tikhonov functionals with general similarity data

<abstract><p>We study a unified form of Tikhonov regularization for ill-posed problems with a general data similarity term. We discuss sufficient conditions on this generalized Tikhonov functional that guarantee existence and stability of solutions. Furthermore, we show that some particular cases of similarity functionals and regularization techniques can be cast into a unified theoretical framework. In particular, we consider the cases of $ p $-power norm, Bregman distance and mutual information as examples of a data similarity term, and Tikhonov regularization of order one or using a Sobolev norm, total variation penalization and powers of semi-norms associated to closed operators as examples of a regularization term. Finally, we take up the case of the image registration problem.</p></abstract>

The index function and Tikhonov regularization for ill-posed problems

In this paper, we study the regularizing properties of the conditional stability estimates in ill-posed problems. First, we analyze how conditional stability estimates occur, and which properties the corresponding index functions must obey. In addition, we adapt the convergence analysis for the Tikhonov regularization in Banach spaces where the difference between the approximated solution and the exact one in metric measure is taken into account. We conclude this study with a comparison of stability estimates and variational inequalities, another emerging tool in Banach space regularization.

Two-parameter discrepancy principle for combined projection and Tikhonov regularization of ill-posed problems

Abstract. For solving a linear ill-posed problem, a combination of Tikhonov regularization and a finite dimensional projection method is considered. In the present paper we treat the dimension of the projection as the second parameter of regularization. The method using the discrepancy set of all regularization parameter pairs which satisfy a certain discrepancy principle is investigated. For the case of truncated SVD and LSQ projection methods, under the standard source conditions, an order of convergence is derived. This order of convergence is optimal under the same conditions as those for Tikhonov regularization with the discrepancy principle and without additional discretization.

Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces

Nonstationary iterated Tikhonov regularization is an efficient method for solving ill-posed problems in Hilbert spaces. However, this method may not produce good results in some situations since it tends to oversmooth solutions and hence destroy special features such as sparsity and discontinuity. By making use of duality mappings and Bregman distance, we propose an extension of this method to the Banach space setting and establish its convergence. We also present numerical simulations which indicate that the method in Banach space setting can produce better results.

A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level

We consider Tikhonov regularization of linear ill-posed problems with noisy data. The choice of the regularization parameter by classical rules, such as discrepancy principle, needs exact noise level information: these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level. We propose a general family of parameter choice rules, which includes many known rules and guarantees convergence of approximations. Quasi-optimality is proved for a sub-family of rules. Many rules from this family work well also in the case of many times under- or overestimated noise level. In the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post-estimated monotone error rule and a certain new rule from the proposed family. The advantages of the new rules are demonstrated in extensive numerical experiments.

Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces

Even if the recent literature on enhanced convergence rates for Tikhonov regularization of ill-posed problems in Banach spaces shows substantial progress, not all factors influencing the best possible convergence rates under sufficiently strong smoothness assumptions were clearly determined. In particular, it was an open problem whether the structure of the residual term can limit the rates. For residual norms of power type in the functional to be minimized for obtaining regularized solutions, the latest rates results for nonlinear problems by Neubauer [On enhanced convergence rates for Tikhonov regularization of non-linear ill-posed problems in Banach spaces, Inverse Prob. 25 (2009), p. 065009] indicate an apparent qualification of the method caused by the residual norm exponent p. The new message of the present article is that optimal rates are shown to be independent of that exponent in the range 1 ≤ p < ∞. However, on the one hand, the smoothness of the image space influences the rates, and on the other hand best possible rates require specific choices of the regularization parameters α > 0. While for all p > 1 the regularization parameters have to decay to zero with some prescribed speed depending on p when the noise level tends to zero in order to obtain the best rates, the limiting case p = 1 shows some curious behaviour. For that case, the α-values must get asymptotically frozen at a fixed positive value characterized by the properties of the solution as the noise level decreases.

On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces

It is well known for the Tikhonov regularization of ill-posed problems in Hilbert spaces that the best possible convergence rate is O(α) in the noise-free case and in the noisy case if the exact solution x† satisfies the source condition . We will show under what conditions the best possible rates with respect to α may be obtained for nonlinear ill-posed problems in Banach spaces. Moreover, we present convergence rate results with respect to the Bregman distance and show special cases where these rates are best possible.

Some tendencies in the Tikhonov regularization of ill-posed problems

Some tendencies in the Tikhonov regularization of ill-posed problems

Some tendencies in the Tikhonov regularization of ill-posed problems

Some tendencies in the Tikhonov regularization of ill-posed problems

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.

Supplement to An A Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates

Supplement to An A Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates

An A Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates

We propose an a posteriori parameter choice for ordinary and iterated Tikhonov regularization that leads to optimal rates of convergence towards the best approximate solution of an ill-posed linear operator equation in the presence of noisy data. Numerical examples are given.

Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates

We propose a class ofa posteriori parameter choice strategies for Tikhonov regularization (including variants of Morozov's and Arcangeli's methods) that lead to optimal convergence rates toward the minimal-norm, least-squares solution of an ill-posed linear operator equation in the presence of noisy data.

An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates

We propose an a posteriori parameter choice for ordinary and iterated Tikhonov regularization that leads to optimal rates of convergence towards the best approximate solution of an ill-posed linear operator equation in the presence of noisy data. Numerical examples are given.