On generalized iterated Tikhonov regularization with operator-dependent seminorms

We investigate the recently introduced Tikhonov regularization filters with penalty terms having seminorms that depend on the operator itself. Exploiting the singular value decomposition of the operator, we provide optimal order conditions, smoothing properties, and a general condition (with a minor condition of the seminorm) for the saturation level. Moreover, we introduce and analyze both stationary and nonstationary iterative counterparts of the generalized Tikhonov method with operator-dependent seminorms. We establish their convergence rate under conditions affecting only the iteration parameters, proving that they overcome the saturation result. Finally, some selected numerical results confirm the effectiveness of the proposed regularization filters.