A proximal regularized Gauss-Newton-Kaczmarz method and its acceleration for nonlinear ill-posed problems

We propose and analyze Kaczmarz-type methods that related to proximal algorithms to solve the nonsmooth hybrid regularization models which are derived from collections of N coupled nonlinear operator equations. To begin with, we introduce a proximal regularized Gauss-Newton-Kaczmarz (PRGNK) method which is constructed by combining the Kaczmarz strategy with a proximal regularized Gauss-Newton (PRGN) iteration. Its convergence analysis is presented under appropriate assumptions, and the numerical experiments on large-scale diffuse optical tomography and parameter identification problems indicate that, PRGNK is clearly faster than the PRGN iteration. Moreover, we incorporate a Nesterov-type acceleration scheme into PRGNK in order to further accelerate the convergence, which leads to a so-called accelerated proximal regularized Gauss-Newton-Kaczmarz (APRGNK) method. Based on the discussion for PRGNK, we also establish the convergence analysis of APRGNK. Meanwhile, the numerical simulations explicitly show that APRGNK makes a remarkable acceleration effect compared with its non-accelerated counterpart. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.