We study multi-parameter Tikhonov regularization, i.e., with multiple penalties. Such models are useful when the sought-for solution exhibits several distinct features simultaneously. Two choice rules, i.e., discrepancy principle and balancing principle, are studied for choosing an appropriate (vector-valued) regularization parameter, and some theoretical results are presented. In particular, t...

A convergence rate is established for nonstationary iterated Tik-honov regularization, applied to ill-posed problems involving closed, densely deened 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.

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of ...

Abstract. In this paper we derive scale space methods for inverse problems which satisfy the fundamental axioms of fidelity and causality and we provide numerical illustrations of the use of such methods in deblurring. These scale space methods are asymptotic formulations of the Tikhonov-Morozov regularization method. The analysis and illustrations relate diffusion filtering methods in image pr...

In the article, Tikhonov regularization of control-constrained optimal control problems is investigated. Typically the solutions of such problems exhibit a so-called bang-bang structure. We develop a parameter choice rule that adaptively selects the Tikhonov regularization parameter depending on a-posteriori computable quantities. We prove that this choice leads to optimal convergence rates wit...

In this paper we discuss general regularization estimators. This class includes Tikhonov type and spectral cut-off estimators as well as iterative methods, such as ν-methods and the Landweber iteration. The latter estimators achieve the same (optimal) convergence rates as spectral cut-off, but do not require explicit spectral information on the operator and are often much faster to compute than...

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization ...

In learning theory, the convergence issues of the regression problem are investigated with the least square Tikhonov regularization schemes in both the RKHS-norm and the L -norm. We consider the multi-penalized least square regularization scheme under the general source condition with the polynomial decay of the eigenvalues of the integral operator. One of the motivation for this work is to dis...

Image restoration is one of the classical inverse problems in image processing and computer vision, which consists in recovering information about the original image from incomplete or degraded data. This paper presents analytical solution for onedimensional case of the Tikhonov regularization method and algorithm of parameter α selection by discrepancy, which finds the mostly smoothed function...