Multilevel augmentation methods for nonlinear ill-posed problems

Abstract

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.

Keywords:

• nonlinear ill-posed problems
• monotone operator
• multilevel augmentation methods

2010 AMS Subject Classification :

• 65J20

Acknowledgements

The authors are grateful to the referees for their constructive comments which has led to the improvement of this paper. This research is partially supported by the Natural Science Foundation of China under grants 10771224 and 11071264, the Science and Technology Section of SINOPEC, Guangdong Provincial Government of China through the ‘Computational Science Innovative Research Team’ programme.