Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation
For researchers in inverse problems, this provides a theoretical foundation for heuristic rules in convex settings, but the extension is incremental.
The paper extends convergence analysis of heuristic parameter choice rules for convex Tikhonov regularisation, proving convergence under noise restrictions similar to the linear case, with examples in ℓ^q spaces.
We investigate the convergence theory of several known as well as new heuristic parameter choice rules for convex Tikhonov regularisation. The success of such methods is dependent on whether certain restrictions on the noise are satisfied. In the linear theory, such conditions are well understood and hold for typically irregular noise. In this paper, we extend the convergence analysis of heuristic rules using noise restrictions to the convex setting and prove convergence of the aforementioned methods therewith. The convergence theory is exemplified for the case of an ill-posed problem with a diagonal forward operator in $\ell^q$ spaces. Numerical examples also provide further insight.