Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces
For researchers in inverse problems and regularization, this provides theoretical optimality guarantees for sparsity-promoting wavelet methods, though the results are incremental extensions of existing variational source condition frameworks.
This paper proves that Tikhonov regularization with wavelet Besov norm penalties achieves minimax-optimal convergence rates for linear and nonlinear ill-posed inverse problems, including the Radon transform and nonlinear differential equation problems, under both deterministic and statistical noise models.
This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on $B^0_{p,1}$ penalty terms 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 it is shown 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.