Vanishing-Regularization Convergence for Nonlinear Iterations

Definitions and mathematical statement

The supplied record asks for an extension of a convergence analysis for the iteration \[ g_{n+1}=g_k+B(S(g_n))-B_k(S(g_n)). \] Its proposed replacement of $S$ is the iteration-dependent composition \[ S_n=N_{\rho,\alpha_n}\circ R, \qquad \alpha_n\longrightarrow 0\quad(n\to\infty), \] where $N_{\rho,\alpha_n}$ is described in the source record as involving Tikhonov regularization. Formally, the resulting iteration is \[ g_{n+1}=g_k+B\left(N_{\rho,\alpha_n}(R(g_n))\right) -B_k\left(N_{\rho,\alpha_n}(R(g_n))\right). \] Requested extension. Can the convergence analysis of the original iteration be extended to this iteration with vanishing regularization parameters?

Unresolved qualifications. The supplied record does not define the spaces, operators $B,B_k,S,R,N_{\rho,\alpha}$, parameter $\rho$, initial data, or the role of the index $k$. In particular, it does not specify whether $k$ is fixed as $n$ varies. It supplies neither the Tikhonov minimization functional and minimizer selection nor the hypotheses and conclusion of the original convergence analysis. The intended limit, topology of convergence, admissible parameter sequences beyond $\alpha_n\to0$, and any noise or boundary assumptions are therefore unspecified. The displayed question preserves the supplied formulation but does not constitute a self-contained convergence conjecture with determined quantifiers and hypotheses.

Short English statement

The question asks whether an existing convergence analysis remains valid when each iteration uses Tikhonov regularization with a parameter tending to zero. The operators, analytic setting, and intended convergence conclusion are missing from the supplied record, so a precise self-contained conjecture requires further source clarification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json: permanent ID $30001995$, source problem code catalogue entry, title Vanishing-Regularization Convergence for Nonlinear Iterations. The supplied record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2012, and names no proposer.
• Supplied report citation: Inverse Problems for Partial Differential Equations (2012), Oberwolfach Reports, DOI 10.4171/owr/2012/11, pp. 611--659. https://doi.org/10.4171/owr/2012/11. The supplied literature-source label is "Oberwolfach Report 11 2012." No precise problem page or named problem author is supplied.
• The supplied statement status is reconstructed_unverified. Its verification note states that the official report confirms the displayed iteration and vanishing-regularization question, while the short excerpt does not define all operators sufficiently for a self-contained formulation. This source-verification claim has not been independently checked here.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. The record labels the problem open and reports that the convergence extension remains open and that no primary paper establishing it in this setting was located. The supplied checked source is https://doi.org/10.4171/owr/2012/11. The record explicitly calls for independent verification of time-sensitive status and cited claims.