  We consider a class of regularization methods for inverse problems where a
coupled regularization is employed for the simultaneous reconstruction of data
from multiple sources. Applications for such a setting can be found in
multi-spectral or multi-modality inverse problems, but also in inverse problems
with dynamic data. We consider this setting in a rather general framework and
derive stability and convergence results, including convergence rates. In
particular, we show how parameter choice strategies adapted to the interplay of
different data channels allow to improve upon convergence rates that would be
obtained by treating all channels equally. Motivated by concrete applications,
our results are obtained under rather general assumptions that allow to include
the Kullback-Leibler divergence as data discrepancy term. To simplify their
application to concrete settings, we further elaborate several practically
relevant special cases in detail. To complement the analytical results, we also
provide an algorithmic framework and source code that allows to solve a class
of jointly regularized inverse problems with any number of data discrepancies.
As concrete applications, we show numerical results for multi-contrast MR and
joint MR-PET reconstruction.
