Impact of conditional stability: Convergence rates for general linear regularization methods

Abstract

The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with an injective, compact linear forward operator A : X → Y mapping between Hilbert spaces X, Y. We are going to combine the ideas of [S. I. Kabanikhin, Convergence rate estimation of gradients methods via conditional stability of inverse and ill-posed problems. J. Inv. Ill-Posed Problems13 (2005), 259–264.] and [S. I. Kabanikhin, Conditional stability stopping rule for gradient methods applied to inverse and ill-posed problems. J. Inv. Ill-Posed Problems14 (2006), 805–812.] concerning the conditional stability on some subset M ⊂ X and the ideas of [B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods. SIAM J. Numer. Anal.45 (2007), 1122–1141.] concerning profile functions in order to find convergence rates for general linear regularization methods. Moreover, an extension to Lavrentiev regularization is given.