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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/125/02/0227-0238

• # Keywords

Tikhonov regularization; nonlinear ill-posed problems; discrepancy principle; 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 Freéchet 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&amp;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 assumption 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&amp;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).

• # Author Affiliations

1. Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019