Performance of Krylov Subspace Methods for Symmetric Matrices in Hybrid Parallelization

We deal with Krylov subspace methods such as the Conjugate Gradient (CG) method for solving linear equations with symmetric matrices on a parallel computer. The algorithm which has the less number of synchronization (abbreviated as synchro) points is crucial for reducing the communication time on the parallel computer. CG has two synchro points per iteration, and the AZMJ variant of Orthomin(2) (abbreviated as AZMJ) which has just one synchro point has been proposed. A number of strategies to generate preconditioners have been known for obtaining successful and rapid convergence. We apply the Symmetric Successive Over Relaxation (SSOR) preconditioner to their methods. Then extra computational costs are required and we need to compute the forward and back substitution in the preconditioned algorithm. We therefore propose an alternative SSOR splitting for the parallel computing, and a computation procedure to parallelize the forward and back substitution and to reduce the computational costs. The numerical results show that the convergence behavior of AZMJ is superior to that of CG, and the parallel performance of AZMJ, which has the less number of synchro points than CG, is higher using the hybrid parallelization on the parallel computer. AZMJ and CG with the preconditioner using our proposed procedure are efficient on the parallel computer, and are useful for obtaining rapid convergence.