Parallel regularized Newton method for nonlinear ill-posed equations

Abstract

We introduce a regularized Newton method coupled with the parallel splitting-up technique for solving nonlinear ill-posed equations with smooth monotone operators. We analyze the convergence of the proposed method and carry out numerical experiments for nonlinear integral equations.Keywords Monotone operator · Regularized Newton method · Parallel splitting-up technique Mathematics Subject Classifications (2010) 47J06 · 47J25 · 65J15 · 65J20 · 65Y05

Parallel methods for regularizing systems of equations involving accretive operators

“…Proof Let x * n be the unique solution of (6). Setting e i n = x i n − x * n ; e n = x n − x * n ; n = α n τ n N , we can rewrite (8) as…”

“…Very recently, several sequential and parallel regularizing methods for solving system (1) have been proposed. The Kaczmarz method, [1,2] the Newton-Kacmarz method, [3] the steepest-descent-Kaczmarz method, [4] parallel iterative regularization methods, [5] parallel regularized Newton-type methods, [6,7] and parallel hybrid methods, [8] to name only few. However, most of the investigation of available methods was carried out in the framework of Hilbert spaces.…”