Heuristic parameter-choice rules for convex variational regularization based on error estimates

TL;DR

This paper develops and analyzes heuristic parameter choice rules for convex variational regularization, extending existing methods from quadratic cases, with theoretical error estimates, convergence analysis, and numerical validation.

Contribution

It introduces generalized heuristic rules for convex regularization, providing error estimates and convergence proofs, expanding beyond quadratic regularization methods.

Findings

Hanke-Raus rule derived with a posteriori error estimates

Quasi-optimality criterion generalized for convex regularization

Numerical experiments demonstrate applicability of both rules

Abstract

In this paper, we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates. Two such rules are derived and generalize those from quadratic regularization, namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error estimates are shown for the Hanke-Raus rule, and convergence for both rules is also discussed. Numerical results for both rules are presented to illustrate their applicability.