  This paper is devoted to proving convergence rates of variational and
iterative regularization methods under variational source conditions VSCs for
inverse problems whose linearization satisfies a range invariance condition. In
order to achieve this, often an appropriate relaxation of the problem needs to
be found that is usually based on an augmentation of the set of unknowns and
leads to a particularly structured reformulation of the inverse problem. We
analyze three approaches that make use of this structure, namely a variational
and a Newton type scheme, whose convergence without rates has already been
established in \cite{rangeinvar}; additionally we propose a split minimization
approach that can be show to satisfy the same rates results. \\ The range
invariance condition has been verified for several coefficient identification
problems for partial differential equations from boundary observations as
relevant in a variety of tomographic imaging modalities. Our motivation
particularly comes from the by now classical inverse problem of electrical
impedance tomography EIT and we study both the original formulation by a
diffusion type equation and its reformulation as a Schr\"odinger equation. For
both of them we find relaxations that can be proven to satisfy the range
invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with
the abstract framework for the three approaches mentioned above, we arrive at
convergence rates results for the variational, split minimization and Newton
type method in EIT.
