On maximum residual nonlinear Kaczmarz-type algorithms for large nonlinear systems of equations

Abstract

Recently, a class of nonlinear Kaczmarz (NK) algorithms has been proposed to solve large-scale nonlinear systems of equations. The NK algorithm is a generalization of the Newton–Raphson (NR) method and does not need to compute the entire Jacobian matrix. In this paper, we present a maximum residual nonlinear Kaczmarz (MRNK) algorithm for solving large-scale nonlinear systems of equations, which employs a maximum violation row selection and acts only on single rows of the entire Jacobian matrix at a time. Furthermore, we also establish the convergence theory of MRNK. In addition, inspired by the effectiveness of block Kaczmarz algorithms for solving linear systems, we further present a block MRNK (MRBNK) algorithm based on an approximate maximum residual criterion. Based on sketch-and-project technique and sketched Newton–Raphson method, we propose the deterministic sketched Newton–Raphson (DSNR) method which is equivalent to MRNBK, and then the global convergence theory of DSNR is established based on some assumptions and μ-strongly quasi-convex condition. Furthermore, the convergence theory of DSNR is provided under star-convex assumption. Finally, some numerical examples are tested to show the effectiveness of our new technique.