Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations

Abstract

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation in Banach space using an a posteriori parameter choice strategy and Morozov-type stopping rule. The method considered by Jin Qinian was an extension of the method proposed by M. Hanke in 1997 for the Hilbert space case. In this paper, we suggest a modified variant of the method, namely, the simplified Levenberg–Marquardt scheme in Banach spaces. The advantage of the method considered in the paper is that, it is also applicable for the operator equation with non-smooth operator. We establish convergence of the method under a modified a posteriori parameter choice strategy which is more feasible than the one considered by Jin Qinian (2011). Numerical example to demonstrate the validity of the considered method is discussed.

Keywords:

• Nonlinear ill-posed operator equations
• Levenberg–Marquardt method in Banach spaces
• Bregman distance
• duality mapping
• Morozov-type stopping rule

2010 Mathematics Subject Classifications:

• 65J20
• 65J22
• 47J06
• 47J25

Acknowledgements

The first author acknowledges Science and Engineering Research Board (SERB), Department of Science & Technology, Government of India for the support for the work through MATRICS project file no. MTR/2019/000670 dated 11 February, 2020.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Science and Engineering Research Board (SERB), Department of Science & Technology, Government of India [MATRICS project file no. MTR/2019/000670].