REGINN-IT method with general convex penalty terms for nonlinear inverse problems

Abstract

In the spirit of the method in Margotti and Rieder (An inexact Newton regularization in Banach spaces based on the non-stationary iterated Tikhonov method. J Inverse Ill-Posed Probl. 2015;23:373–392), we propose an inexact Newton regularization which is based on the non-stationary iterated Tikhonov method for non-smooth solutions of nonlinear inverse problems in Banach spaces. The method consists of an inner iteration and an outer iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method. The inner iteration is performed by the non-stationary iterated Tikhonov method. The remarkable point is that the penalty terms can be general convex functions. Under certain assumptions on nonlinear operators, the convergence analysis of our method is given by making use of tools from convex analysis. Furthermore, numerical simulations are provided to support the theoretical results and test the performance of the method.

Keywords:

• Inexact Newton regularization
• non-stationary iterated Tikhonov method
• nonlinear inverse problems
• general convex penalty terms

2010 Mathematics Subject Classifications:

• 35R30
• 65L09

Acknowledgments

The authors would like to thank Dr Qinian Jin (Australian National University, Australia) for some useful comments. The work is supported by the National Natural Science Foundation of China (No.11871180) and the Fundamental Research Funds for the Central Universities.

Disclosure statement

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

Additional information

Funding

The work is supported by the National Natural Science Foundation of China [grant number 11871180] and the Fundamental Research Funds for the Central Universities.