数学
雅可比矩阵与行列式
非线性系统
牛顿法
残余物
趋同(经济学)
算法
应用数学
局部收敛
一般化
基质(化学分析)
迭代法
数学优化
数学分析
物理
量子力学
材料科学
经济
复合材料
经济增长
作者
Jianhua Zhang,Yuqing Wang,Jing Zhao
标识
DOI:10.1016/j.cam.2023.115065
摘要
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.
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