Using Landweber method to quantify source conditions - a numerical study
For practitioners of inverse problems, this work provides a numerical method to estimate source condition parameters, which are typically unverifiable, enabling data-independent parameter choice rules.
The authors show that source conditions in inverse problems imply a Kurdyka-Łojasiewicz inequality, and use the Landweber method to numerically approximate the source condition parameter μ. Numerical experiments demonstrate that the implied lower bound on convergence rate is of optimal order and observable without knowing μ.
Source conditions of the type $x^†\in\mathcal{R}((A^\ast A)^μ)$ 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-Łojasiewicz inequality with certain parameters depending on $μ$. While the converse implication is unclear from a theoretical point of view, we demonstrate how the Landweber method in combination with the Kurdyka-Łojasiewicz inequality can be used to approximate $μ$ 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 $μ$.