Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability

Abstract

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.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

“…u † ∈ X 1 (note, however, that this is not explicitly required anywhere), and the regularization error u δ α − u † is still measured in the norm of X. This continues former studies like [9] under the assumption u † ∈ X p for some 0 < p < 1. In contrast to those papers, the focus of the present work is, although also not explicitly required anywhere, on the case u † ∈ X p for each 0 < p < 1, consequently on the situation characterized by p = 0.…”

“…In the present work, we discuss the nonlinear Tikhonov regularization (1.3) in particular with an oversmoothing penalty term, where we have u † ∈ X 1 = D(B), or in other words u † 1 = +∞. This continues studies started in papers [12,11] and [9], where convergence rates and numerical case studies are provided for a priori and a posteriori parameter choices, respectively, under certain smoothness assumptions on u † and structural conditions on F . Under the same structural conditions, which are also similar to those in the corresponding seminal paper for linear operator equations by Natterer [20], we present as the novelty of this paper convergence results based on the Banach-Steinhaus theorem without needing any smoothness assumptions.…”

“…In the following assumption, we briefly summarize the structural properties of the operator F and of its domain D(F ), in particular with respect to the the solution u † of equation (1.3). For examples of nonlinear inverse problems, which satisfy these assumptions (or at least substantial parts of it), we refer to [6,9] and to the appendices of the papers [11,28]. ASSUMPTION 2.1.…”

The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

“…for all x ∈ D (6) and constants 0 < c a ≤ C a < ∞. The left-hand inequality in Equation (6) represents a conditional stability estimate and is substantial for obtaining stable regularized solutions, whereas the right-hand inequality in Equation (6) contributes to the determination of the nonlinearity structure of the forward operator F. Convergence and rate results for the Tikhonov regularization expressed in Equation (4) with oversmoothing penalties under the inequality chain expressed in Equation (6) were recently presented in [3,6,9] and complemented by case studies in [10]. The present paper continues this series of articles by addressing open questions with respect to the discrepancy principle for choosing the regularization parameter α and its comparison to a priori parameter choices.…”

“…The present paper continues this series of articles by addressing open questions with respect to the discrepancy principle for choosing the regularization parameter α and its comparison to a priori parameter choices. In this context, one of the examples from [10] is reused for performing new numerical experiments in order to obtain additional assertions that cannot be taken from analytical investigations. The paper is organized as follows: We summarize in Section 2 basic properties of regularized solutions under assumptions that are typical for oversmoothing penalties and in Section 3 assertions concerning the convergence.…”