Lavrentiev's regularization method in Hilbert spaces revisited

Abstract

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

Optimal rates for Lavrentiev regularization with adjoint source conditions

“…Hofmann, Kaltenbacher, and Resmerita [8] studied Lavrentiev regularization under variational source conditions. These authors did not explicitly focus on the bias, instead they showed the convergence rate for the overall regularization error.…”

“…The authors in [1] and [8] prove convergence results u − u δ γ → 0 as δ → 0, both for a priori and a posteriori parameter choices, under the additional assumption that the solution u of (1) is a minimum-norm solution. For elements u which satisfy a source condition, either direct or adjoint, this automatically holds true, because u is in the orthogonal complement of the nullspace N (A).…”

“…In order to better understand the character of the range type source conditions (8) and (9) from Definition 4 we mention a technical lemma which formulates a relation between the ranges of the fractional powers of accretive linear operators A and A * . Lemma 1.…”

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

“…First our focus is on the Lavrentiev regularization approach based on formula (2.13), in which κ(j δ , g δ ) ∈ L 2 (Ω). The general theory of linear Lavrentiev regularization (see, e.g., [43] and also [2,12,24,36,37]) yields convergence and convergence rates results for the error of regularized solutions f ρ,δ with respect to the uniquely determined f * -minimizing solution f † to problem (IP −M N ). Taking into account that Lemma 2.2 holds, we immediately derive (see, e.g., [12,Rem.…”

“…Due to formula (1.18), the Tikhonov regularization approach under consideration with specific misfit term also appears as a variant of the Lavrentiev regularization (see, e.g., [2,12,24,43]). After some operatortheoretic settings and preliminary results in Section 2, concerning also the ill-posedness of the linear inverse problem under consideration, we apply in Section 3 the general theory of classical Tikhonov and Lavrentiev regularization for such problems yielding propositions on convergence and convergence rates for the regularized solutions in the infinite dimensional Hilbert spaces.…”