Proposal of a Genetic Algorithm-applied GMRES(m) method with Automatic Subspace Parameter Optimization

Abstract

This paper presents an approach for improving the efficiency of solving linear systems by applying a genetic algorithm (GA) to the GMRES(m) method. For every restart process in GMRES(m), the initial vectors are regarded as chromosomes. When the restart process stagnates, the GA performs a crossover on chromosomes to create new chromosomes for the next restart stage, in which a weighted average algorithm is used to perform the crossover process effectively. To further enhance the performance, the concept of “chromosome-wide stagnation” is introduced by enabling on-the-fly detection of a slowdown in convergence of the GA. A possible way to adjust the m value automatically at the onset of such stagnation is proposed. The proposed method had been tested on several sample matrices and showed satisfactory improvements in execution time.