Convergence rates of convex variational regularization

The aim of this paper is to provide quantitative estimates for the minimizers of non-quadratic regularization problems in terms of the regularization parameter, respectively the noise level. As usual for ill-posed inverse problems, these estimates can be obtained only under additional smoothness assumptions on the data, the so-called source conditions, which we identify with the existence of Lagrange multipliers for a limit problem. Under such a source condition, we shall prove a quantitative estimate for the Bregman distance induced by the regularization functional, which turns out to be the natural distance measure to use in this case. We put a special emphasis on the case of total variation regularization, which is probably the most important and prominent example in this class. We discuss the source condition for this case in detail and verify that it still allows discontinuities in the solution, while imposing some regularity on its level sets.