Preconditioned iterative regularization in Banach spaces

Regularization methods for inverse problems formulated in Hilbert spaces usually give rise to over-smoothness, which does not allow to obtain a good contrast and localization of the edges in the context of image restoration. On the other hand, regularization methods recently introduced in Banach spaces allow in general to obtain better localization and restoration of the discontinuities or localized impulsive signals in imaging applications. We present here an expository survey of the topic focused on the iterative Landweber method. In addition, preconditioning techniques previously proposed for Hilbert spaces are extended to the Banach setting and numerically tested.