Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

Abstract

We study the choice of the regularization 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 analyze 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 analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.

Keywords:

• Generalized cross-validation
• ill-posed problems
• inverse problems
• large noise
• parameter choice rules
• regularization

Acknowledgment

The authors thank the reviewers for their valuable input which helped to improve the rigor as well as the presentation of the article.

Additional information

Funding

This work was supported by the Austrian Science Fund (FWF) project P 30157-N31.