The Iterated Fractional Tikhonov Regularization For Ill-posed Problems

Fractional Tikhonov regularization methods have been recently proposed to reduce the over smoothing 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,non-stationary iterated fractional Tikhonov regularization methods are investigated,establishing their convergence rate under general conditions on the iteration parameters.