A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations

Abstract

In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y; where the right hand side is replaced by noisy data y? ? X with ?y - y ?? ? ? and T : D(T) ? X ? X is a nonlinear monotone operator defined on a Hilbert space X: The iteration x ?<inf>n,?</inf> converges quadratically to the unique solution x<inf>?</inf>? of the equation T(x) + ?(x - x<inf>0</inf>) = y? (x<inf>0</inf> := x <inf>0,?</inf>?). It is known that (Tautanhahn (2002)) x<inf>?</inf>? converges to the solution x? of Tx = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x <inf>0</inf> - x? we proved that the error ?x? - x <inf>n, ?</inf>?;? is of optimal order. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate. © 2012 Institute of Mathematics, NAS of Belarus.