A distributed and parallel unite and conquer method to solve sequences of non-Hermitian linear systems

Abstract

Many problems in science and engineering often require to solve a long sequence of large-scale non-Hermitian linear systems with different right-hand sides (RHSs) but a unique operator. Efficiently solving such problems on extreme-scale platforms requires the minimization of global communications, reduction of synchronization and promotion of asynchronous communications. Unite and Conquer GMRES/LS-ERAM (UCGLE) method (Wu and Petiton, in Proceedings of the International Conference on High Performance Computing in Asia-Pacific Region. ACM, New York, pp 36–46, https://doi.org/10.1145/3149457.3154481 , 2018) is a suitable candidate with the reduction of global communications and the synchronization points of all computing units. It consists of three computing algorithms with asynchronous communications that allow the use of approximated eigenvalues to accelerate the convergence of solving linear systems and to improve fault tolerance. In this paper, we extend both the mathematical model and the implementation of UCGLE method to adapt to solve sequences of linear systems. The eigenvalues obtained in solving previous linear systems by UCGLE can be recycled, improved on the fly and applied to construct a new initial guess vector for subsequent linear systems, which can achieve a continuous acceleration to solve linear systems in sequence. Numerical experiments using different test matrices to solve sequences of linear systems on supercomputer Tianhe-2 indicate a substantial decrease in both computation time and iteration steps when the approximated eigenvalues are recycled to generate the initial guess vectors.