A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems with Multiple Right-hand Sides

Xinzhe Wu (1, 2, 3) , Serge Petiton (1, 2, 3)

1 CNRS - Centre National de la Recherche Scientifique
2 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
3 MDLS - Maison de la Simulation

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 the Unite and Conquer GMRES/LS-ERAM (UCGLE) method [1] by combining it 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 allow the use of approximate eigenvalues to accelerate to solve of linear systems and to improve their fault tolerance. In this paper, the variant of UCGLE with novel components and manager engine implementations is 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 and reusability. 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.

Mots clés: Linear systems; Krylov subspace methods; unite and conquer; multiple right-hand sides; asynchronous communication

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Analyse numérique [cs.NA]

Xinzhe Wu, Serge Petiton. A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems with Multiple Right-hand Sides. Parallel Computing, 2019, pp.102551. ⟨10.1016/j.parco.2019.102551⟩. ⟨hal-01918741⟩