Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term

The nonstationary preconditioned iteration proposed in a recent work by Donatelli and Hanke appeared on IP can be seen as an approximated iterated Tikhonov method. Starting from this observation we extend the previous iteration in two directions: the introduction of a regularization operator different from the identity (e.g., a differential operator) and the projection into a convex set (e.g., the nonnegative cone). Depending on the application both generalizations can lead to an improvement in the quality of the computed approximations. Convergence results and regularization properties of the proposed iterations are proved. Finally, the new methods are applied to image deblurring problems and compared with the iteration in the original work and other methods with similar properties recently proposed in the literature.