Generalized Tikhonov regularization and modern convergence rate theory in Banach spaces

The development of regularization methods for solving ill-posed inverse problems started decades ago, but is still a major topic in applied mathematics. One prominent example is the method of Tikhonov introduced in the nineteen sixties. Till the late eighties research focused on Tikhonov regularization for linear equations in Hilbert spaces. Only during the last 25 years also nonlinear Problems were considered and after the turn of the millenium also Banach space settings were taken into account. Motivated by applications in imaging new and generalized versions of Tikhonov’s method arose and the theoretic basis had to be widened appropriately. Different concepts for investigating the behavior of such new approaches were suggest by several authors. For people only interested in the theoretic foundations of Tikhonov-type methods but not doing research in this field things thus became very complex. In this monograph we present and analyze a very general formulation of Tikhonov’s method which covers almost all Tikhonov-type approaches given in the literature. An extensive example shows how to apply the findings to a highly topical problem in imaging. We also investigate the relations between several modern concepts for obaining convergence rates, which are of major interest in regularization theory. Our results show that many ideas which have evolved during the last years can be unified ending up with only two different flavors of one concept.