Using Landweber iteration to quantify source conditions – a numerical study
Abstract Source conditions of the type {x^{\dagger}\in\mathcal{R}((A^{\ast}A)^{\mu})} are a standard assumption 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–Łojasiewicz inequality with certain parameters depending on μ. While the converse implication is unclear from a theoretical point of view, we propose an algorithm which represents a first attempt that allows to approximate the value of μ numerically. It is based on combining the Landweber iteration with the Kurdyka–Łojasiewicz inequality. We conduct several numerical experiments to demonstrate the potential and limitations of the current method. 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 μ.