Heuristic Parameter Choice Rules for Tikhonov Regularization with 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.

Displacement field estimation from OCT images utilizing speckle information with applications in quantitative elastography

“…In case the noise level δ is unknown or estimates of it are unreliable, then instead of the discrepancy principle one can use heuristic stopping rules such as the heuristic discrepancy principle [23,24], which determines the stopping index k * by minimizing…”

Displacement field estimation from OCT images utilizing speckle information with applications in quantitative elastography

“…In case the noise level δ is unknown or estimates of it are unreliable, then instead of the discrepancy principle one can use heuristic stopping rules such as the heuristic discrepancy principle [23,24], which determines the stopping index k * by minimizing…”

Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation

“…We omit the proof as it is analogous to the above. Note that if the source condition (10) holds, α * is selected according to the right quasi-optimality rule and the auto-regularisation condition (27) is satisfied, then one may also prove that…”

“…It is important to note that the aforementioned noise conditions utilised the spectral theory for self-adjoint linear operators. A recent discussion and extension of the noise conditions within the linear theory may be found in [27].…”