** Smoothness concepts and convergence rates for Tikhonov-type regularization in Banach spaces**

The monograph by Engl, Hanke and Neubauer presented 17 years ago the fundamentals of a systematic theory for the regularization of inverse problems in Hilbert spaces, including convergence rates for the Tikhonov-type regularization of nonlinear ill-posed operator equations. In general, error estimates and rate results with respect the Hilbert space norm for nonlinear ill-posed problems are based on two ingredients: (A) structural conditions concerning the nonlinearity of the forward operator and (B) source conditions expressing the smoothness of the solution with respect to the forward operator. In the context of an extension to Banach spaces, alternative error measures like the Bregman distance became popular in the past ten years, and we have introduced approximate source conditions also for the Banach space setting. Variational inequalities of speciﬁc type (also called variational source conditions) became of interest, in particular for the treatment of non-smooth situations, since they combine solution smoothness and nonlinearity conditions in a sophisticated manner. We show that in the case of l¹-regularization such variational inequalities also exist if source conditions are not applicable, because sparsity constraints are narrowly missed, but a decay asymptotics of the solution components is known. For that case a curious form of solution smoothness, coming from properties of the exploited Schauder basis, plays an important role. Some cross connections between variational inequalities and conditional stability estimates occurring in inverse PDE problems can be veriﬁed. The theory was applied, for example, to inverse option pricing problems and autoconvolution problems in ultrashort laser pulse characterization.