Simplified Levenberg–Marquardt Method in Hilbert Spaces

Abstract

In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ ( x ) = y , where F : D ⁢ ( F ) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as

Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ ( x 0 ) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ ( F ) for the exact solution x † and ( F ′ ⁢ ( x 0 ) ) * denote the adjoint of F ′ ⁢ ( x 0 ) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method.