  Fractional Tikhonov regularization methods have been recently proposed 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. This paper provides saturation and
converse results on their convergence rates. Using the same iterative
refinement strategy of iterated Tikhonov regularization, new iterated
fractional Tikhonov regularization methods are introduced. We show that these
iterated methods are of optimal order and overcome the previous saturation
results. Furthermore, nonstationary iterated fractional Tikhonov regularization
methods are investigated, establishing their convergence rate under general
conditions on the iteration parameters. Numerical results confirm the
effectiveness of the proposed regularization iterations.
