Schwarz Methods for the Time-Parallel Solution of Parabolic Control Problems

  • Martin J. Gander
  • Felix KwokEmail author
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Discretized parabolic control problems lead to very large systems of equations, because trajectories must be approximated forward and backward in time. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? We present a convergence analysis for a class of Schwarz methods applied to a model parabolic optimal control problem. We show that just applying a classical Schwarz method in time already implies better transmission conditions than the ones usually used in the elliptic case, and we propose an even better variant based on optimized Schwarz theory.


Classical and optimized Schwarz methods Domain decomposition Parabolic control problems 


  1. 1.
    A.T. Barker, M. Stoll, Domain decomposition in time for PDE-constrained optimization (2014, submitted)Google Scholar
  2. 2.
    A. Borzì, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, vol. 8 (SIAM, Philadelphia, 2012)zbMATHGoogle Scholar
  3. 3.
    H.S. Dollar, N.I. Gould, M. Stoll, A.J. Wathen, Preconditioning saddle-point systems with applications in optimization. SIAM J. Sci. Comput. 32(1), 249–270 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    X. Du, M. Sarkis, C.E. Schaerer, D.B. Szyld, Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems. Electron. Trans. Numer. Anal. 40, 36–57 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Gander, F. Kwok, G. Wanner, Constrained Optimization: From Lagrangian Mechanics to Optimal Control and PDE Constraints, in Optimization with PDE Constraints (Springer, New York, 2014), pp. 151–202zbMATHGoogle Scholar
  7. 7.
    E. Haber, A parallel method for large scale time domain electromagnetic inverse problems. Appl. Numer. Math. 58(4), 422–434 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. J. Comput. Appl. Math. 173(1), 169–198 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GenevaGenevaSwitzerland
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloonHong Kong

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