Semi-heuristic parameter choice rules for Tikhonov regularisation with operator perturbations

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.

Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation

“…Proof. From the estimate (12), one may then immediately estimate the data propagation error courtesy of the auto-regularisation condition as…”

“…Note that from Proposition 4 and since α * is the global minimiser, for any α, we have that α * ≤ Cψ HD (α * , y δ ) ≤ Cψ HD (α, y δ ). Observe that from (12), it follows that…”

“…Let α * be the minimiser of the Hanke-Raus functional. Then, as before, we estimate the Bregman distance as (12) and from (22), deduce that…”

A new heuristic parameter choice rule in Tikhonov regularization applied for Ritz approximation of an ill-posed problem

“…The study of heuristic methods of selecting the regularization parameter is still reasonable (see [14,15,12,5]) because of the importance and frequency of situations where no delta estimation is available.…”

“…This holds for A compact. Recently, the use of a lower bound for α in a semiheuristic rule has also been considered in [5].…”