An Adaptive Quantile-Based Block Kaczmarz Method for Nonlinear Equations

Abstract This paper introduces an adaptive block Kaczmarz algorithm for solving large-scale systems of nonlinear equations. The core innovation is a quantile-based criterion for dynamically selecting the working rows at each iteration. By focusing on equations associated with the largest residuals, the method efficiently projects the iterate onto the corresponding hyperplanes, circumventing the need for computationally expensive submatrix pseudoinverses. This approach offers enhanced flexibility in block size selection compared to existing block nonlinear Kaczmarz methods. We provide a rigorous convergence analysis, demonstrating that with an appropriate quantile parameter, the upper bound on the convergence rate is superior to that of related Kaczmarz-type algorithms. Numerical experiments on a variety of test problems confirm that the proposed method significantly outperforms its counterparts in both iteration count and computational time. Mathematics Subject Classification: 65H10, 65F10, 65Y20