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

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. [ABSTRACT FROM AUTHOR]