Scalable hierarchical parallel algorithm for the solution of super large-scale sparse linear equations

TL;DR

This paper introduces a hierarchical parallel algorithm for efficiently solving super large-scale sparse linear equations, demonstrating high scalability and efficiency on a supercomputer with over a billion degrees of freedom.

Contribution

A novel hierarchical parallel master-slave iterative algorithm that reduces communication overhead and accelerates convergence for large-scale sparse linear systems.

Findings

Achieved efficient solution of billion-degree systems on 2001 processors.

Demonstrated high parallel efficiency and scalability.

Applicable to large-scale implicit finite element analyses.

Abstract

The parallel linear equations solver capable of effectively using 1000+ processors becomes the bottleneck of large-scale implicit engineering simulations. In this paper, we present a new hierarchical parallel master-slave-structural iterative algorithm for the solution of super large-scale sparse linear equations in distributed memory computer cluster. Through alternatively performing global equilibrium computation and local relaxation, our proposed algorithm will reach the specific accuracy requirement in a few of iterative steps. Moreover, each set/slave-processor majorly communicate with its nearest neighbors, and the transferring data between sets/slave-processors and master is always far below the set-neighbor communication. The corresponding algorithm for implicit finite element analysis has been implemented based on MPI library, and a super large 2-dimension square system of triangle-lattice truss structure under random static loads is simulated with over one billion degrees of freedom and up to 2001 processors on "Exploration 100" cluster in Tsinghua University. The numerical experiments demonstrate that this algorithm has excellent parallel efficiency and high scalability, and it may have broad application in other implicit simulations.