# Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation

## TL;DR

This paper extends the convergence analysis of heuristic parameter choice rules for convex Tikhonov regularisation, demonstrating their effectiveness under noise restrictions and providing theoretical and numerical insights.

## Contribution

It introduces new convergence results for heuristic rules in convex Tikhonov regularisation, expanding the linear theory to the convex setting with practical examples.

## Key findings

- Convergence of heuristic rules under noise restrictions is established.
- Numerical examples illustrate the theoretical results.
- Analysis applies to ill-posed problems with diagonal operators in ^q spaces.

## Abstract

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.