Discrepancy principle for generalized GN iterations combined with the reverse connection control

Abstract

Abstract In this paper we investigate the generalized Gauss–Newton method in the following form x n+1 = ξn – θ(F′*(xn )F′(xn ), τn )F′*(xn ){(F(xn ) – ƒ δ ) – F′(xn )(xn – ξn )}, x 0, ξn ∈ 𝒟 ⊂ H 1. The modified source condition which depends on the current iteration point xn , is used. We call this inclusion the undetermined reverse connection. The new source condition leads to a much larger set of admissible control elements ξn as compared to the previously studied versions, where ξn = ξ. The process is combined with a novel a posteriori stopping rule, where is the number of the first transition of ‖F(xn ) – ƒ δ ‖ through the given level δω , 0, < ω < 1, i.e., The convergence analysis of the proposed algorithm is given.