Quadratic convergence of Levenberg-Marquardt method for general nonlinear inverse problems with two parameters

Abstract

In this paper, we shall present the quadratic convergence of Levenberg-Marquardt (L-M) method for solving general ill-posed nonlinear inverse problems including two parameters to be identified. As the Tikhonov regularized minimizations are non-convex due to the nonlinearity of the nonlinear inverse problems, the L-M method transforms them into convex minimizations. The quadratic convergence of the L-M method is rigorously demonstrated under some basic assumptions. An application of the L-M method for the simultaneous identification of the Robin coefficient and heat flux in an elliptic system is presented, and the surrogate functional technique is introduced to solve the convex but still strongly ill-conditioned minimizations, resulting in an explicit minimizer of the minimization at each L-M iteration. Numerical experiments are presented to illustrate the efficiency and accuracy of the proposed algorithm.

Keywords:

• Inverse problems
• Levenberg-Marquardt method
• quadratic convergence
• ill-posedness
• surrogate functional

2010 Mathematics Subject Classifications:

• 65M32
• 65M12
• 90C25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Ting Cheng and Daijun Jiang was financially supported by National Natural Science Foundation of China (Nos. 11871240, 11771170, 11401241 and 11571265) and Fundamental Research Funds for the Central Universities CCNU19TD010. The work of Daijun Jiang was also financially supported by NSFC-RGC (China-Hong Kong, No. 11661161017).