An iteratively regularized projection method with quadratic convergence for nonlinear Ill-posed problems

Abstract

An iteratively regularized projection method, which converges quadratically, has been considered for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x) = y where F : D(F) ? X ? X is a nonlinear monotone operator defined on the real Hilbert space X: We assume that only a noisy data y? with y-y? ? ? are available. Under the assumption that the Fréchet derivative F? of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that under a general source condition on x<inf>0</inf> - x?, the error between the regularized approximation where P<inf>h</inf> is an orthog-onal projection on to a nite dimensional subspace X<inf>h</inf> of X) and the solution x? is of optimal order.