Simplified Levenberg-Marquardt Method in Hilbert Spaces

摘要

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) subset of 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)(delta)} is defined as x(n+1)(delta) = x(n)(delta) (alpha I-n + F'(x(0))* F'(x(0)))(-1) F'(x(0))* (F(x(n)(delta)) - y(delta)). Here {alpha(n)} is a decreasing sequence of nonnegative numbers which converges to zero, F'(x(0)) denotes the Frechet derivative of F at an initial guess x(0) is an element of D(F) for the exact solution x(dagger) and F'(x(0)))* denote the adjoint of F'(x(0)). In our proposed method, we need to calculate Frechet 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.