Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence-rates

Abstract

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equation Tx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution x-cap belongs to the range of the operator (T*T)sup(v), 0 less-than-or-equal-to 1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate if v=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2 less-than-or-equal-to v less-than-or-equal-to 1.