Hanke-Raus heuristic rule for variational regularization in Banach spaces
This work extends a known heuristic parameter selection method to a broader class of problems (Banach spaces), but the results are theoretical and incremental in nature.
The paper generalizes the Hanke-Raus heuristic parameter choice rule to variational regularization for ill-posed inverse problems in Banach spaces, proving convergence results with and without source conditions, and demonstrating performance with numerical examples.
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.