We report on joint work with Philip Miller on Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on Besov norms with fine index 1, which yield estimators that are sparse with respect to a wavelet frame. Our framework includes, among others, the Radon transform and some nonlinear inverse problems in differential equations with distributed measurements.
Using variational source conditions we show that such estimators achieve minimax-optimal rates of convergence for finitely smoothing operators in certain Besov balls both for deterministic and for statistical noise models.
Using explicit bounds on the number of nonvanishing coefficients of the estimator, we also derive a converse result for approximation rates.