  We consider a modified Tikhonov-type functional for the solution of ill-posed
nonlinear inverse problems. Motivated by applications in the field of
production engineering, we allow small deviations in the solution, which are
modeled through a tolerance measure in the regularization term of the
functional. The existence, stability, and weak convergence of minimizers are
proved for such a functional, as well as the convergence rates in the Bregman
distance. We present an example for illustrating the effect of tolerances on
the regularized solution and examine parameter choice rules for finding the
optimal regularization parameter for the assumed tolerance value. In addition,
we discuss the prospect of reconstructing sparse solutions when tolerances are
incorporated in the regularization functional.
