Morozov-type discrepancy principle for nonlinear ill-posed problems under η-condition

Abstract

For proving the existence of a regularization parameter under a Morozov- type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freechet derivative and requirement of the Lipschitz constant to depend on a source condition is one such restriction (Ramlau P, Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). Another nonlinearity condition considered by Scherzer (Computing, 51 (1993) 45-60) was by requiring the forward operator to be close to a linear operator in a restricted sense. A seemingly natural nonlinear assump- tion which appears in many applications which attracted attention in various contexts of the study of nonlinear problems is the so-called η-condition. However, a Morozov-type discrepancy principle together with η-condition does not seem to have been studied, except in a recent paper by the author (Bull. Aust. Math. Soc. 79 (2009) 337-342), where error estimates under a general source condition is derived, by assuming the existence of the parameter. In this paper, the existence of the parameter satisfying a Morozov-type discrepancy principle is proved under the η-condition on the forward operator, by assuming the source condition as in the papers of Scherzer (Computing, 51 (1993) 45-60) and Ramlau (Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). This source condition is, in fact, a special case of the source condition in the author's paper (Bull. Aust. Math. Soc. 79 (2009) 337-342).