On nonlinear greedy randomized Kaczmarz methods for solving large-scale systems of nonlinear equations

Abstract

The greedy randomized Kaczmarz (GRK) method is an efficient method for solving large-scale systems of linear equations. Based on this basic idea, we propose a nonlinear greedy randomized Kacz-marz (NGRK) method for solving large-scale systems of nonlinear equations and prove its convergence. Then, to improve the computational efficiency of the NGRK method, we design the multi-step NGRK (MNGRK) method and the accelerated NGK (ANGK) method, respectively. Theoretical results show that the convergence speed of the MNGRK and ANGK methods is faster than that of the NGRK method. Finally, inspired by the max-distance coordinate descent (MDCD) method, we also establish the nonlinear max-distance Kaczmarz (NMDK) method and its accelerated version.