Penalty-based smoothness conditions in convex variational regularization
This work provides a theoretical framework for error analysis in convex variational regularization, benefiting researchers in inverse problems and regularization theory.
The authors propose new smoothness conditions for convex variational regularization of linear ill-posed problems, which yield error bounds for Bregman distances that split into a smoothness-dependent term and a noise-dependent term, similar to standard quadratic Tikhonov regularization in Hilbert spaces.
The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.