Variational Regularization Theory Based on Image Space Approximation Rates

Abstract: We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for $0,2,q$- and $0,p,p$-penalties without restrictions on $p,q\in (1,\infty).$ Finally we prove equivalence of H\"older-type variational source conditions, bounds on the defect of the Tikhonov functional, and image space approximation rates.