Solving Linear Equations Using a Jacobi Based Time-Variant Adaptive Hybrid Evolutionary Algorithm

TL;DR

This paper introduces a novel Jacobi-based time-variant adaptive hybrid evolutionary algorithm for solving large sparse linear equations, demonstrating improved convergence speed and effectiveness over existing hybrid and classical methods.

Contribution

The paper proposes a new TVA-based adaptive relaxation factor technique integrated with Jacobi methods, enhancing convergence and local tuning in hybrid evolutionary algorithms.

Findings

Outperforms JBUA hybrid algorithm in convergence speed

Achieves better effectiveness than classical methods

Theoretically proven convergence of the proposed algorithm

Abstract

Large set of linear equations, especially for sparse and structured coefficient (matrix) equations, solutions using classical methods become arduous. And evolutionary algorithms have mostly been used to solve various optimization and learning problems. Recently, hybridization of classical methods (Jacobi method and Gauss-Seidel method) with evolutionary computation techniques have successfully been applied in linear equation solving. In the both above hybrid evolutionary methods, uniform adaptation (UA) techniques are used to adapt relaxation factor. In this paper, a new Jacobi Based Time-Variant Adaptive (JBTVA) hybrid evolutionary algorithm is proposed. In this algorithm, a Time-Variant Adaptive (TVA) technique of relaxation factor is introduced aiming at both improving the fine local tuning and reducing the disadvantage of uniform adaptation of relaxation factors. This algorithm integrates the Jacobi based SR method with time variant adaptive evolutionary algorithm. The convergence theorems of the proposed algorithm are proved theoretically. And the performance of the proposed algorithm is compared with JBUA hybrid evolutionary algorithm and classical methods in the experimental domain. The proposed algorithm outperforms both the JBUA hybrid algorithm and classical methods in terms of convergence speed and effectiveness.