A distributed and parallel asynchronous unite and conquer method to solve large scale non-Hermitian linear systems with multiple right-hand sides

Abstract

Many problems in the field of science and engineering often require to solve simultaneously large-scale non-Hermitian sparse linear systems with multiple right-hand sides (RHSs). Efficiently solving such problems on extreme-scale platforms requires the minimization of global communications, reduction of synchronization points and promotion of asynchronous communications. We develop an extension of Unite and Conquer GMRES/LS-ERAM (UCGLE) method [1] by replacing its GMRES Component with Block GMRES method to solve non-Hermitian linear systems with multiple RHSs. UCGLE is a hybrid method consisting of three computing algorithms with asynchronous communication that allows the use of approximated eigenvalues to accelerate the convergence of the solves of linear systems and to improve their fault tolerance. In this paper, the variant of UCGLE with new components and manager engine implementation are introduced. This engine is capable of allocating multiple Block GMRES at the same time, each Block GMRES solving the linear systems with a subset of RHSs and accelerating the convergence using the eigenvalues approximated by other eigensolvers. Dividing the entire linear system with multiple RHSs into subsets and solving them simultaneously with different allocated linear solvers allow localizing calculations, reducing global communication, and improving parallel performance. Meanwhile, the asynchronous preconditioning using eigenvalues is able to speed up the convergence and improve the fault tolerance. Numerical experiments using different test matrices on supercomputer ROMEO indicate that the proposed method achieves a substantial decrease in both computation time and iterative steps with good scaling performance.