On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems

In the past few years, convergence rates results for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces have been published, where the classical concept of source conditions was replaced with variational inequalities holding on some level sets. This essentially advanced the analysis of non-smooth situations with respect to forward operators and solutions. In fact, such variational inequalities combine both structural conditions on the nonlinearity of the operator and smoothness properties of the solution. Varying exponents in the variational inequalities correspond to different levels of convergence rates. In this article, we discuss the range of occurring exponents in the Banach space setting. To lighten the cross-connections between generalized source conditions, degree of nonlinearity of the forward operator and associated variational inequalities we study the Hilbert space situation and even prove some converse result for linear operators. Finally, we outline some aspects for the interplay of variational regularization and conditional stability estimates for partial differential equations. As an example, we apply the theory to a specific parameter identification problem for a parabolic equation.