Sourcewise representation of solutions to nonlinear operator equations in a Banach space and convergence rate estimates for a regularized Newton’s method

Abstract

We present a class of iterative methods for solving nonlinear ill-posed operator equations with differentiable operators in a Banach space. The methods are based on regularization of the linearized equation by using an appropriate regularization scheme at each iteration. We establish that the iterations converge locally with power rate provided that the solution admits of a sourcewise representation. We also prove that the condition of sourcewise representability is very close to a necessary condition for this kind of estimates.

Convergence and application of a modified iteratively regularized Gauss–Newton algorithm

“…The iteratively Gauss-Newton method (IRGN) for the solution of exponentially ill-posed inverse problems, which was introduced by Bakushinsky in 1993 [BA93], has been investigated and extended in a number of formulations, [BK01,BK04,BS05,BKA06,K00]. Here, our goal is to extend the IRGN algorithm to nonlinear problems in which physical considerations suggest replacement of the stabilizing term of the IRGN by a more general Tikhonov penalty stabilizing term.…”

Inverse problem in optical tomography and its numerical investigation by iteratively regularized methods

“…For the convergence analysis of IRGN and MGNM we follow [5,6], where the parametric approximations formalism was developed to construct iterative processes for nonlinear irregular operator equations in Hilbert spaces: For the convergence analysis of IRGN and MGNM we follow [5,6], where the parametric approximations formalism was developed to construct iterative processes for nonlinear irregular operator equations in Hilbert spaces:…”

“…In Section 4 we present the computational algorithms and state the main convergence theorem following [5,6]. A novel a posteriori stopping rule for these methods is suggested.…”