Case studies and a pitfall for nonlinear variational regularization under conditional stability
For researchers in inverse problems, this work clarifies conditions for reliable regularization under conditional stability, highlighting a critical pitfall that can invalidate results.
This paper analyzes Tikhonov regularization under conditional stability for nonlinear ill-posed problems, proving convergence rates for oversmoothing penalties and demonstrating that incorrect stability estimates can lead to complete failure of convergence.
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this paper we consider Tikhonov regularization under conditional stability estimates for nonlinear ill-posed operator equations in Hilbert scales. We summarize assertions on convergence and convergence rate in three cases describing the relative smoothness of the penalty in the Tikhonov functional and of the exact solution. For oversmoothing penalties, for which the rue solution no longer attains a finite value, we present a result with modified assumptions for a priori choices of the regularization parameter yielding convergence rates of optimal order for noisy data. We strongly highlight the local character of the conditional stability estimate and demonstrate that pitfalls may occur through incorrect stability estimates. Then convergence can completely fail and the stabilizing effect of conditional stability may be lost. Comprehensive numerical case studies for some nonlinear examples illustrate such effects.