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An iteratively regularized projection method, which converges quadratically, is considered for stable approximate solutions to a nonlinear ill-posed operator equation 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 yδ with ‖y−yδ ‖ ≤ δ 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 x0− x̂, the error ‖xh,δ n,α − x̂‖ between the regularized approximation x h,δ n,α , (x h,δ 0,α := Phx0, where Ph is an orthogonal projection on to a finite dimensional subspace Xh of X) and the solution x̂ is of optimal order.

Expanding the applicability of an iterative regularization method for ill-posed problems