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

Abstract

For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Hölder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies, without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

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

“…From Assumption 1, we have that there are no solutions x * ∈ D = D(F) ∩ D(B), satisfying with F(x * ) = y the operator expressed in Equation (1), because this would contradict, with F(x † ) = F(x * ) and x † − x * X > 0, the left-hand inequality of Equation (6). Besides x † , however, other solutions with x * / ∈ D(B) may exist.…”

“…In this section, we discuss assertions about the X-norm convergence of regularized solutions with the Tikhonov functional T δ α introduced in Equation (4). First we recall the following lemma (from [6], Proposition 3.4).…”

“…However, it is well known that there are nonlinear problems, where that version is problematic due to duality gaps that prevent the solvability of Equation (5). For overcoming the remaining weaknesses of the parameter choice expressed in Equation (5), sequential versions of the discrepancy principle can be applied that approximate α discr , and we refer to [4][5][6] for more details. Such an approach is used for performing the numerical case studies in Section 6.…”

“…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 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. In Section 4 we show that the error estimates derived in [6] for obtaining low order convergence rates are also applicable to obtain the order optimal Hölder convergence rates under the associated Hölder-type source conditions. Section 5 recalls a nonlinear inverse problem from an exponential growth model and an appropriate Hilbert scale, which can both be used for performing numerical experiments in the subsequent section.…”