Convergence Rates for Inverse Problems with Impulsive Noise

We study inverse problems $F(f) =g$ with perturbed right-hand side $g^{\rm obs}$ corrupted by so-called impulsive noise, i.e., noise which is concentrated on a small subset of the domain of definition of $g$. It is well known that Tikhonov-type regularization with an $\mathbf{L}^1$ data fidelity term yields significantly more accurate results than Tikhonov regularization with classical $\mathbf{L}^2$ data fidelity terms for this type of noise. The purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy. Our error estimates significantly improve previous error estimates for Tikhonov regularization with $\mathbf{L}^1$-fidelity term in the case of impulsive noise. We present numerical results which are in good agreement with the predictions of our analysis.