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

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 mu-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.(c) 2023 Elsevier B.V. All rights reserved.