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Asynchronous iterations of HSS method for non-Hermitian linear systems (2312.16505v1)

Published 27 Dec 2023 in math.NA, cs.DC, and cs.NA

Abstract: A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for $\mathsf H$-matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems respectively arising from convection-diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

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References (46)
  1. Z.-Z. Bai. On the convergence of additive and multiplicative splitting iterations for systems of linear equations. J. Comput. Appl. Math., 154(1):195–214, 2003.
  2. Z.-Z. Bai. Regularized HSS iteration methods for stabilized saddle-point problems. IMA J. Numer. Anal., 39(4):1888–1923, 2019.
  3. Modified HSS iteration methods for a class of complex symmetric linear systems. Computing, 87(3):93–111, 2010.
  4. Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput., 28(2):583–603, 2006.
  5. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 24(3):603–626, 2003.
  6. Z.-Z. Bai and M. Rozložník. On the numerical behavior of matrix splitting iteration methods for solving linear systems. SIAM J. Numer. Anal., 53(4):1716–1737, 2015.
  7. G. M. Baudet. Asynchronous iterative methods for multiprocessors. J. ACM, 25(2):226–244, 1978.
  8. Asynchronous iterative algorithms with flexible communication for nonlinear network flow problems. J. Parallel Distrib. Comput., 38(1):1 – 15, 1996.
  9. M. Benzi. A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl., 31(2):360–374, 2009.
  10. M. Benzi and D. Bertaccini. Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal., 28(3):598–618, 2008.
  11. M. Benzi and J. Liu. An efficient solver for the incompressible Navier-Stokes equations in rotation form. SIAM J. Sci. Comput., 29(5):1959–1981, 2007.
  12. M. Benzi and D. B. Szyld. Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numer. Math., 76(3):309–321, 1997.
  13. Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. Numer. Math., 99(3):441–484, 2005.
  14. D. P. Bertsekas. Distributed asynchronous computation of fixed points. Math. Program., 27(1):107–120, 1983.
  15. Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1989.
  16. D. Chazan and W. Miranker. Chaotic relaxation. Linear Algebra Appl., 2(2):199–222, 1969.
  17. V. Conrad and Y. Wallach. Alternating methods for sets of linear equations. Numer. Math., 32(1):105–108, 1979.
  18. MPI for Python. J. Parallel Distrib. Comput., 65(9):1108–1115, 2005.
  19. J. Douglas. On the numerical integration of ∂2u∂x2+∂2u∂y2=∂u∂tsuperscript2𝑢superscript𝑥2superscript2𝑢superscript𝑦2𝑢𝑡\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=% \frac{\partial u}{\partial t}divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u end_ARG start_ARG ∂ italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG ∂ italic_u end_ARG start_ARG ∂ italic_t end_ARG by implicit methods. J. Soc. Ind. Appl. Math., 3(1):42–65, 1955.
  20. Synchronous and asynchronous optimized Schwarz methods for one-way subdivision of bounded domains. Numer. Linear Algebra Appl., 27(2):e2227, 2020.
  21. M. N. El Tarazi. Some convergence results for asynchronous algorithms. Numer. Math., 39(3):325–340, 1982. (in French).
  22. K. Fan. Topological proofs for certain theorems on matrices with non-negative elements. Monatshefte für Mathematik, 62:219–237, 1958.
  23. A. Frommer and D. B. Szyld. H-splittings and two-stage iterative methods. Numer. Math., 63(1):345–356, 1992.
  24. A. Frommer and D. B. Szyld. Asynchronous two-stage iterative methods. Numer. Math., 69(2):141–153, 1994.
  25. A. Frommer and D. B. Szyld. Asynchronous iterations with flexible communication for linear systems. Calculateurs Parallèles, 10:421–429, 1998.
  26. G. Gbikpi-Benissan and F. Magoulès. Protocol-free asynchronous iterations termination. Adv. Eng. Softw., 146:102827, 2020.
  27. M. R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6):409–436, 1952.
  28. Y.-M. Huang. A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math., 255:142–149, 2014.
  29. A single-step HSS method for non-Hermitian positive definite linear systems. Appl. Math. Lett., 44:26–29, 2015.
  30. Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Linear Algebra Appl., 14(3):217–235, 2007.
  31. F. Magoulès and G. Gbikpi-Benissan. JACK: An asynchronous communication kernel library for iterative algorithms. J. Supercomput., 73(8):3468–3487, 2017.
  32. F. Magoulès and G. Gbikpi-Benissan. Asynchronous Parareal time discretization for partial differential equations. SIAM J. Sci. Comput., 40(6):C704–C725, 2018.
  33. 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.
  34. 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.
  35. Asynchronous iterations of Parareal algorithm for option pricing models. Mathematics, 6(4):1–18, 2018.
  36. Asynchronous optimized Schwarz methods with and without overlap. Numer. Math., 137(1):199–227, 2017.
  37. F. Magoulès and C. Venet. Asynchronous iterative sub-structuring methods. Math. Comput. Simul., 145:34–49, 2018.
  38. G. I. Marchuk. Splitting and alternating direction methods. In Handbook of Numerical Analysis, volume 1, pages 197–462. Elsevier, 1990.
  39. J.-C. Miellou. Algorithmes de relaxation chaotique à retards. ESAIM: M2AN, 9(R1):55–82, 1975. (in French).
  40. The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math., 3(1):28–41, 1955.
  41. Y. Saad and M. H. Schultz. Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7(3):856–869, 1986.
  42. S. Schechter. Relaxation methods for linear equations. Comm. Pure Appl. Math., 12(2):313–335, 1959.
  43. J. W. Sheldon. On the numerical solution of elliptic difference equations. MTAC, 9(51):101–112, 1955.
  44. S.-L. Wu. Several variants of the Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems. Numer. Linear Algebra Appl., 22(2):338–356, 2015.
  45. Performance of asynchronous optimized Schwarz with one-sided communication. Parallel Comput., 86:66–81, 2019.
  46. Q. Zou and F. Magoulès. Parameter estimation in the Hermitian and skew-Hermitian splitting method using gradient iterations. Numer. Linear Algebra Appl., 27:e2304, 2020.
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