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

  • Xinzhe WuEmail author
  • Serge G. Petiton
Special Feature: Original Paper International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2018)


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,, 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.


Sequence of linear systems Linear solvers Krylov methods Unite and Conquer Iterative methods Eigenvalues 

Mathematics Subject Classification

00A69 65F10 



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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Maison de la SimulationGif-sur-Yvette CedexFrance
  2. 2.CRIStAL, Université de LilleVilleneuve d’AscqFrance

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