Noise Conditions for Quasioptimal Total-Variation Regularization

Definitions and mathematical statement

Total-variation regularization denotes regularization using a total-variation penalty. The supplied record does not specify the underlying domain, function spaces, forward operator, data-fidelity functional, or definition of total variation. Generalized quasioptimality and Hanke--Raus are the names of the parameter-choice rules in the record; their defining functionals and parameter-selection conventions are not supplied. Let $N$ denote the sought noise-restriction set, whose ambient space and admissibility conditions remain unspecified.

Source question. Determine a noise-restriction set $N$ ensuring convergence of total-variation regularization with generalized quasioptimality and Hanke--Raus parameter-choice rules, described in the source as the "combined method."

Unresolved formulation. The record does not determine whether the two rules are applied separately or combined into a single procedure. It also does not specify the noise model, the meaning of vanishing noise, the reconstruction-selection convention, the convergence topology, or the intended limiting solution. In particular, it does not state whether convergence is required for every admissible noise sequence, every solution, or every regularized minimizer. Consequently, a fully quantified convergence assertion and a precise definition of the requested set $N$ cannot be extracted from the supplied record without additional hypotheses.

Short English statement

The question asks which restrictions on noise guarantee convergence of total-variation regularization using generalized quasioptimality and Hanke--Raus parameter choices. The supplied record leaves the regularization setting, the relationship between the rules, and the meaning of convergence unspecified, so the intended theorem requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30002212$, exact source problem code catalogue entry; title: "Noise Conditions for Quasioptimal Total-Variation Regularization." No individual proposer is identified; the proposed year is recorded as 2012.
• Supplied source citation: Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory (2013), Oberwolfach Reports, pp. 3061--3127, DOI 10.4171/owr/2012/51. Supplied source URL and literature-source label "original OWR problem": https://doi.org/10.4171/owr/2012/51. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The connection between this report title and the regularization question requires source review.
• Supplied literature source, labelled "noise-level-free convergence analysis": https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol38/abstract.php?pages=233-257.
• Supplied literature source, labelled "Hanke--Raus variational regularization": https://arxiv.org/abs/1606.00115.
• Supplied dated literature curation, checked at 2026-08-21: status "partially solved." Its assessment states that convergence results exist for quasioptimality and Hanke--Raus rules under deterministic-noise, source, or Banach-space hypotheses, but that no general noise-restriction set ensuring convergence for total-variation regularization has been established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the cited official report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification assertions, not verification performed for this draft.