  We address the classical issue of appropriate choice of the regularization
and discretization level for the Tikhonov regularization of an inverse problem
with imperfectly measured data. We focus on the fact that the proper choice of
the discretization level in the domain together with the regularization
parameter is a key feature in adequate regularization. We propose a
discrepancy-based choice for these quantities by applying a relaxed version of
Morozov's discrepancy principle. Indeed, we prove the existence of the
discretization level and the regularization parameter satisfying such
discrepancy. We also prove associated regularizing properties concerning the
Tikhonov minimizers.
