Heuristic Parameter Choice Rules for Tikhonov Regularisation with Weakly Bounded Noise
For researchers in inverse problems, this paper provides theoretical guarantees for parameter selection under weaker noise assumptions, but the results are incremental extensions of existing methods.
This paper studies heuristic parameter choice rules for Tikhonov regularization of linear ill-posed problems with weakly bounded noise and unknown noise level, proving convergence and convergence rates for adapted quasi-optimality, heuristic discrepancy, and Hanke-Raus rules, and providing conditions for generalized cross-validation and predictive mean-square error rules.
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.