Iterated fractional Tikhonov regularization

Abstract

We consider linear operator equations of the formwhere K : X → Y is a compact linear operator between Hilbert spaces X and Y. We assume y to be attainable, i.e., that problem (1) has a solution x † = K † y of minimal norm. Here K † denotes the (Moore-Penrose) generalized inverse operator of K, which is unbounded when K is compact, with infinite dimensional range. Hence problem (1) is ill-posed and has to be regularized in order to compute a numerical solution. We want to approximate the solution x † of the equation (1) In this paper, we firstly provide a saturation result similar to the well-known saturation result for Tikhonov regularization [6]: indeed, Tikhonov regularization under suitable a-priori assumption and a-priori choice rule, α = α(δ) ∼ c(δ) 2/3 , is of optimal order and the best possible convergence rate obtainable is). On the other hand, let R(K) be the range of K and let Q be the orthogonal projector onto R(K), ifis not closed, and this shows how Tikhonov regularization for an ill-posed problem with compact operator never yields a convergence rate which is faster than O(δ 2 3 ), since it saturates at this rate. Such results motivated us to introduce the iterated versions of fractional Tikhonov methods in the same spirit of the iterated Tikhonov method. We prove that those iterated methods can overcome the afore-mentioned saturation results.Afterwards, inspired by the works [4,5] we introduce the nonstationary variants of our iterated methods. Differently from the nonstationary iterated Tikhonov, we have two nonstationary sequences of parameters. In the noise free case, we give sufficient conditions on these sequences to guarantee the convergence providing also the corresponding convergence rates. In the noise case, we show the stability of the proposed iterative schemes proving that they are regularization methods. For the brief space we have at disposal, we will only cite some of the main results we obtained for one kind of fractional Tikhonov algorithms, that we call weighted Tikhonov. For the results about the other fractional Tikhonov algorithm analysis, proofs, numerical examples and full details, we invite the interested reader to look at the full paper [7].

Iterated Tikhonov regularization with a general penalty term

“…Available analyses of iterated Tikhonov regularization only treat the case when L is the identity, that is, the iteration in Equation . However, computed results reported by Huang et al showed that iterative application of Equation with L ≠ I can give better approximations of x † than Equation .…”