Hanke-Raus heuristic rule for variational regularization in Banach spaces

TL;DR

This paper extends the Hanke-Raus heuristic rule to variational regularization in Banach spaces, providing error estimates and convergence results for solving ill-posed inverse problems, supported by numerical experiments.

Contribution

It generalizes the heuristic parameter choice rule to Banach spaces and derives convergence results under various conditions, including source conditions and noise assumptions.

Findings

Error estimates in Bregman distance are established.

Four convergence results are proven under different assumptions.

Numerical experiments demonstrate the method's effectiveness.

Abstract

We generalize the heuristic parameter choice rule of Hanke-Raus for quadratic regularization to general variational regularization for solving linear as well as nonlinear ill-posed inverse problems in Banach spaces. Under source conditions formulated as variational inequalities, we obtain a posteriori error estimates in term of Bregman distance. By imposing certain conditions on the random noise, we establish four convergence results; one relies on the source conditions and the other three do not depend on any source conditions. Numerical results are presented to illustrate the performance.