Iterated fractional Tikhonov regularization
Authors: Bianchi Davide , Buccini Alessandro , Donatelli Marco , Serra‐Capizzano Stefano 
Published: October 2015

We consider linear operator equations of the form

where
$K:{\cal X}\to{\cal Y}$
is a compact linear operator between Hilbert spaces
${\cal X} \hbox{ and } {\cal Y}.$
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),
when only an approximation yδ of y is available with ‖yδ − y‖ ≤ δ, where δ is called the noise level. Since K†yδ is not a good approximation of x†, we approximate x† with xδα := Rαyδ where {Rα} is a family of continuous operators depending on a parameter α. A classical example is the Tikhonov regularization defined by Rα = (K*K + αI)−1 K*, where I denotes the identity and K* the adjoint of K.

Recently, new Tikhonov based regularization methods have been proposed in [1], [2] and [3], under the name of fractional Tikhonov, to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order.

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 ‖xδα−x†‖ = O(δ2/3). On the other hand, let R(K) be the range of K and let Q be the orthogonal projector onto
$\overline{R(K)},$,
if

then x† = 0, as long as R(K) is 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(δ2/3), 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]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)