Random Greedy Fast Block Kaczmarz Method for Solving Large-Scale Nonlinear Systems

TL;DR

This paper introduces a novel Random Greedy Fast Block Kaczmarz method that efficiently solves large-scale nonlinear systems by combining random and greedy strategies, achieving linear convergence without expensive computations.

Contribution

The paper presents a new algorithm that improves large-scale nonlinear system solving by avoiding pseudoinversion and providing theoretical convergence guarantees.

Findings

Method achieves linear convergence in expectation.

Performance is significantly improved with favorable stochastic greedy condition number.

Outperforms comparable algorithms in efficiency and robustness.

Abstract

To efficiently solve large scale nonlinear systems, we propose a novel Random Greedy Fast Block Kaczmarz method. This approach integrates the strengths of random and greedy strategies while avoiding the computationally expensive pseudoinversion of Jacobian submatrices, thus enabling efficient solutions for large scale problems. Our theoretical analysis establishes that the proposed method achieves linear convergence in expectation, with its convergence rates upper bound determined by the stochastic greedy condition number and the relaxation parameter. Numerical experiments confirm that when the Jacobian matrix exhibits a favorable stochastic greedy condition number and an appropriate relaxation parameter is selected, the algorithm convergence is significantly accelerated. As a result, the proposed method outperforms other comparable algorithms in both efficiency and robustness.