Heuristic Parameter Choice Rules for Tikhonov Regularisation with Weakly Bounded Noise

TL;DR

This paper investigates heuristic methods for selecting regularisation parameters in ill-posed linear problems affected by weakly bounded noise, providing convergence analysis and adapting existing rules to this challenging noise setting.

Contribution

It introduces adaptations of heuristic parameter choice rules for weakly bounded noise and proves their convergence and rates under specific conditions.

Findings

Convergence of quasi-optimality, heuristic discrepancy, and Hanke-Raus rules established.

Conditions for convergence of generalized cross-validation and predictive mean-square error rules provided.

Adaptations improve parameter choice in the presence of weakly bounded noise.

Abstract

We study the choice of the regularisation parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyse several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyse and provide conditions for the convergence of the parameter choice by the generalised cross-validation and predictive mean-square error rules.