An Improved Levenberg-Marquardt Iteration Method Of Non-linear Ill-posed Problems In Banach Spaces

In this paper, we mainly study the convergence of an improved Levenberg-Marquardt iteration method for the non-linear ill-posed problems in Banach spaces: where x0δ∈D((?)J)∩D(F) and ζ0δ∈(?)J(x0) are an initial guess. J is a proper, weakly lower semi-continuous and uniformly convex function, F is weakly closed, F’(x) denotes the Frechet derivative of F at x∈D(F), F’(x)*denotes the adjoint operator of F’(x). And we give detailed convergence analyses for noise-free data and noisy data.In Chapter1, we give a brief review about the iteration method of nonlinear ill-posed problems, and present the main result of this paper. In Chapter2, we give the basic definition of an improved Levenberg-Marquardt iteration method. Under the relevant constraints, we get the convergence of Levenberg-Marquardt iteration method for noisy data about Bregman distance which is proved in the third chapter. In Chapter3, we prove the convergence for noisy data which is based on the proof of the convergence on noise-free data.