Semi-Heuristic Parameter Choice Rules for Tikhonov Regularisation with Operator Perturbations

TL;DR

This paper introduces semi-heuristic parameter choice rules for Tikhonov regularisation in ill-posed problems with operator perturbations, improving parameter selection when data noise levels are unknown.

Contribution

The paper proposes a new family of semi-heuristic rules for regularisation parameter choice, with proven convergence and demonstrated numerical improvements over standard heuristics.

Findings

New semi-heuristic rules improve parameter selection accuracy.

Convergence of the proposed rules is theoretically established.

Numerical experiments show better performance than existing heuristic methods.

Abstract

We study the choice of the regularisation parameter for linear ill-posed problems in the presence of data noise and operator perturbations, for which a bound on the operator error is known but the data noise-level is unknown. We introduce a new family of semi-heuristic parameter choice rules that can be used in the stated scenario. We prove convergence of the new rules and provide numerical experiments that indicate an improvement compared to standard heuristic rules.