Collective Communication Optimization for Solving Linear Algebraic Equations

With the development of the electronic technology, the processors count in a supercomputer reaches million scales. However, the processes scale of a application is limited to several thousands, and the scalability face a bottle neck from several aspects, including I/O, communication, cache access .etc. In this paper, we focus on the communication bottleneck to the scalability of linear algebraic equation solve. We take preconditioned conjugate gradient (PCG) as an example, and analysis the feathers of the communication operations in the process of PCG solver. We find that reduce communication is the most critical issue for the scalability of the parallel iterative method for linear algebraic equation solve. We propose a local residual error optimization scheme to eliminate part of the reduce communication operations in the parallel iterative method, and improve the scalability of the parallel iterative method. Experimental results on the Tianhe-2 supercomputer demonstrate that our optimization scheme can achieve a much signally effect for the scalability of the linear algebraic equation solve.