Using Landweber method to quantify source conditions - a numerical study

Source conditions of the type $x^\dag \in\mathcal{R}((A^\ast A)^\mu)$ are an important tool in the theory of inverse problems to show convergence rates of regularized solutions as the noise in the data goes to zero. Unfortunately, it is rarely possible to verify these conditions in practice, rendering data-independent parameter choice rules unfeasible. In this paper we show that such a source condition implies a Kurdyka-\L{}ojasiewicz inequality with certain parameters depending on $\mu$. While the converse implication is unclear from a theoretical point of view, we demonstrate how the Landweber method in combination with the Kurdyka-\L{}ojasiewicz inequality can be used to approximate $\mu$ and conduct several numerical experiments. We also show that the source condition implies a lower bound on the convergence rate which is of optimal order and observable without the knowledge of $\mu$.