A Direct Solver for Time Parallelization

  • Martin J. Gander
  • Laurence Halpern
  • Juliet Ryan
  • Thuy Thi Bich Tran
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

With the advent of very large scale parallel computer, having millions of processing cores, it has become important to also use the time direction for parallelization. Among the successful methods doing this are the parareal algorithm, the paraexp algorithm, PFASST and also waveform relaxation methods of Schwarz or Dirichlet-Neumann or Neumann type. We present here a mathematical analysis of a further method to parallelize in time, proposed by Maday and Ronquist in 2007. It is based on the diagonalization of the time stepping matrix. Like for many time domain parallelization methods, this seems at first not to be a very promising approach, since this matrix is essentially triangular, and for a fixed time step even a Jordan block, and thus not diagonalizable. If one however chooses different time steps, diagonalization is possible, and one has to trade of between the accuracy due to necessarily having different time steps, and numerical errors in the diagonalization process of these almost not diagonalizable matrices. We study this trade-off mathematically and propose an optimization strategy for the choice of the parameters, for a Backward Euler discretization of the heat equation in two dimensions.

References

  1. 1.
    A.J. Christlieb, C.B. Macdonald, B.W. Ong, Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    M.J. Gander, S. Güttel, Paraexp: a parallel integrator for linear initial-value problems. SIAM J. Sci. Comput. 35(2), C123–C142 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M.J. Gander, E. Hairer, Nonlinear convergence analysis for the parareal algorithm, in Domain Decomposition Methods in Science and Engineering XVII, ed. by O.B. Widlund, D.E. Keyes, vol. 60 (Springer, Berlin, 2008), pp. 45–56Google Scholar
  4. 4.
    M.J. Gander, L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing. Math. Comput. 74, 153–176 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M.J. Gander, L. Halpern, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 4th edn. Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University Press, Baltimore, 2013)Google Scholar
  7. 7.
    J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics. University Lecture Series, vol. 41 (American Mathematical Society, Providence, 2008)Google Scholar
  8. 8.
    Y. Maday, E.M. Rønquist, Fast tensor product solvers. Part II: spectral discretization in space and time. Technical report 7–9, Laboratoire Jacques-Louis Lions (2007)Google Scholar
  9. 9.
    Y. Maday, E.M. Rønquist, Parallelization in time through tensor-product space-time solvers. C. R. Math. Acad. Sci. Paris 346(1–2), 113–118 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M.L. Minion, A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Martin J. Gander
    • 1
  • Laurence Halpern
    • 2
  • Juliet Ryan
    • 3
  • Thuy Thi Bich Tran
    • 2
  1. 1.University of GenevaGenèveSwitzerland
  2. 2.Université Paris 13VilletaneuseFrance
  3. 3.ONERAChatillonFrance

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