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

Abstract

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 y? with ky? y? k ? ? are available. Under the assumption that the Fréchet derivative F0 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 ? x, the error kx<inf>n</inf> h <inf>?</inf> ? ? xk between the regularized approximation x<inf>n</inf> h <inf>?</inf> ? , (x<inf>0</inf> h <inf>?</inf> ? := P<inf>h</inf>x0, where P<inf>h</inf> is an orthogonal projection on to a finite dimensional subspace X<inf>h</inf> of X) and the solution x is of optimal order. © 2019 Journal of Nonlinear and Variational Analysis