Convergence Rates for Inverse Problems with Impulsive Noise

Resumo/Abstract: We study inverse problems F (f) = g where the data g is corrupted by so-called impulsive noise ξ which is concentrated on a small part of the observation domain. Such noise occurs for example in digital image acquisition. To reconstruct f from noisy measurements we use Tikhonov regularization where it is well-known from numerical studies that L-data fitting yields much better reconstructions than classical L-data fitting. Nevertheless, so far rates of convergence are known only if ‖ξ‖L1 → 0, which does not fully explain the remarkable quality of the reconstructions obtained by L-data fitting. We introduce a continuous model for impulsive noise depending on an impulsiveness parameter η > 0 and prove convergence rates as η → 0. We therefore use a recently developed variational formulation of the noise level and derive expressions for it in terms of η. It turns out that the rates of convergence (compared to the state of the art) clearly improve depending also on the smoothing properties of the forward operator F . Finally we present numerical results, which suggest that our results are order optimal or at least close to order optimal.