Using Landweber method to quantify source conditions - a numerical study

TL;DR

This paper explores how the Landweber method can numerically estimate source conditions in inverse problems, linking them to Kurdyka-ojasiewicz inequalities and demonstrating practical approximation of key parameters.

Contribution

It establishes a connection between source conditions and Kurdyka-ojasiewicz inequalities, and demonstrates how to numerically approximate source condition parameters using the Landweber method.

Findings

The Landweber method can approximate source condition parameters.

Source conditions imply optimal convergence rate bounds.

Numerical experiments validate the practical applicability of the approach.

Abstract

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 $\mu$.