Performance Analysis of the Chebyshev Basis Conjugate Gradient Method on the K Computer

The conjugate gradient (CG) method is useful for solving large and sparse linear systems. It has been pointed out that collective communication needed for calculating inner products becomes serious performance bottleneck when executing the CG method on massively parallel systems. Recently, the Chebyshev basis CG (CBCG) method, a communication avoiding variant of the CG method, has been proposed, and theoretical studies have shown promising results, particularly for upcoming exascale supercomputers. In this paper, we evaluate the CBCG method on an actual system, namely the K computer, to examine the potential of the CBCG method. We first construct a realistic performance model that reflects the computation on the K computer, and the model indicates that the CBCG method is faster than CG method if the number of cores is sufficient large. We then measure the execution time of both methods on the K computer, and obtained results agree with our estimation.