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Resilient asynchronous primal Schur method (2312.14715v1)

Published 22 Dec 2023 in math.NA and cs.NA

Abstract: This paper introduces the application of the asynchronous iterations theory within the framework of the primal Schur domain decomposition method. A suitable relaxation scheme is designed, which asynchronous convergence is established under classical spectral radius conditions. For the usual case where the local Schur complement matrices are not constructed, suitable splittings only based on explicitly generated matrices are provided. Numerical experiments are conducted on a supercomputer for both Poisson's and linear elasticity problems. The asynchronous Schur solver outperformed the classical conjugate-gradient-based one in case of compute node failures.

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References (25)
  1. G. M. Amdahl. Validity of the single processor approach to achieving large scale computing capabilities. In Proceedings of the April 18-20, 1967, Spring Joint Computer Conference, Atlantic City, New Jersey, USA, AFIPS ’67 (Spring), pages 483–485, New York, NY, USA, 1967. ACM.
  2. O. Axelsson and L. Kolotilina. Diagonally compensated reduction and related preconditioning methods. Numer. Linear Algebra Appl., 1(2):155–177, 1994.
  3. Parallel treatment of a class of differential-algebraic systems. SIAM J. Numer. Anal., 33(5):1969–1980, 1996.
  4. G. M. Baudet. Asynchronous iterative methods for multiprocessors. J. ACM, 25(2):226–244, 1978.
  5. Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1989.
  6. Some aspects of parallel and distributed iterative algorithms – A survey. Automatica J. IFAC, 27(1):3–21, 1991.
  7. Convergence of non-stationary parallel multisplitting methods for hermitian positive definite matrices. Math. Comp., 67(221):209–220, 1998.
  8. D. Chazan and W. Miranker. Chaotic relaxation. Linear Algebra Appl., 2(2):199–222, 1969.
  9. K. Fan. Topological proofs for certain theorems on matrices with non-negative elements. Monatshefte für Mathematik, 62:219–237, 1958.
  10. K. Fan. Note on M𝑀Mitalic_M-matrices. Q. J. Math., 11(1):43–49, 1960.
  11. Asynchronous weighted additive Schwarz methods. Electron. Trans. Numer. Anal., 5:48–61, 1997.
  12. A. Frommer and D. B. Szyld. H-splittings and two-stage iterative methods. Numer. Math., 63(1):345–356, 1992.
  13. G. Gbikpi-Benissan and F. Magoulès. Protocol-free asynchronous iterations termination. Adv. Eng. Softw., 146:102827, 2020.
  14. Five types of ruled helical surfaces for helical conveyers, support anchors and screws. MATEC Web Conf., 95:06002, 2017.
  15. F. Magoulès and G. Gbikpi-Benissan. Asynchronous parareal time discretization for partial differential equations. SIAM J. Sci. Comput., 40(6):C704–C725, 2018.
  16. F. Magoulès and G. Gbikpi-Benissan. Distributed convergence detection based on global residual error under asynchronous iterations. IEEE Trans. Parallel Distrib. Syst., 29(4):819–829, 2018.
  17. F. Magoulès and G. Gbikpi-Benissan. JACK2: An MPI-based communication library with non-blocking synchronization for asynchronous iterations. Adv. Eng. Softw., 119:116–133, 2018.
  18. Optimal discrete transmission conditions for a nonoverlapping domain decomposition method for the Helmholtz equation. SIAM J. Sci. Comput., 25(5):1497–1515, 2004.
  19. Asynchronous optimized Schwarz methods with and without overlap. Numer. Math., 137(1):199–227, 2017.
  20. F. Magoulès and C. Venet. Asynchronous iterative sub-structuring methods. Math. Comput. Simulation, 145(Supplement C):34–49, 2018.
  21. S. Schechter. Relaxation methods for linear equations. Comm. Pure Appl. Math., 12(2):313–335, 1959.
  22. Asynchronous Schwarz alternating method with flexible communication for the obstacle problem. Rés. Syst. Répartis Calculateurs Parallèles, 13(1):47–66, 2001.
  23. Parallel asynchronous Schwarz and multisplitting methods for a nonlinear diffusion problem. Numer. Algorithms, 33(1):461–474, 2003.
  24. R. S. Varga. Matrix Iterative Analysis, volume 27 of Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, 2000.
  25. Convergence of two-stage iterative methods using incomplete factorization. J. Comput. Appl. Math., 166(2):565–580, 2004.
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