Advertisement

Efficient Solvers for Some Classes of Time-Periodic Eddy Current Optimal Control Problems

  • Michael Kolmbauer
  • Ulrich LangerEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 45)

Abstract

In this paper, we present and discuss the results of our numerical studies of preconditioned MinRes methods for solving the optimality systems arising from the multiharmonic finite element approximations to time-periodic eddy current optimal control problems in different settings including different observation and control regions, different tracking terms, as well as box constraints for the Fourier coefficients of the state and the control. These numerical studies confirm the theoretical results published by the first author in a recent paper.

Keywords

Time-periodic eddy current optimal control problems Multiharmonic finite element discretization MinRes solver Preconditioners 

Notes

Acknowledgements

The authors gratefully acknowledge the financial support by the Austrian Science Fund (FWF) under the grants P19255 and W1214 (project DK04). The authors also thank the Austria Center of Competence in Mechatronics (ACCM), which is a part of the COMET K2 program of the Austrian Government, for supporting their work on eddy current problems.

References

  1. 1.
    Abbeloos, D., Diehl, M., Hinze, M., Vandewalle, S.: Nested multigrid methods for time-periodic, parabolic optimal control problems. Comput. Visual. Sci. 14(1), 27–38 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Davis, T.A.: Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)Google Scholar
  3. 3.
    Davis, T.A.: A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 165–195 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Davis, T.A., Duff, I.S.: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25, 1–20 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in \(\mathbf{H}(\mathbf{curl})\) and \(\mathbf{H}(\mathrm{div})\) spaces. SIAM J. Numer. Anal. 45(6), 2483–2509 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kolev, T.V., Vassilevski, P.S.: Parallel auxiliary space AMG for \(H(\mathrm{curl})\) problems. J. Comput. Math. 27(5), 604–623 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kollmann, M., Kolmbauer, M.: A preconditioned MinRes solver for time-periodic parabolic optimal control problems. Numer. Lin. Algebra Appl. (2012). doi: 10.1002/nla.1842Google Scholar
  9. 9.
    Kollmann, M., Kolmbauer, M., Langer, U., Wolfmayr, M., Zulehner, W.: A finite element solver for a multiharmonic parabolic optimal control problem. Comput. Math. Appl. 65(3), 469–486 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kolmbauer, M.: The multiharmonic finite element and boundary element method for simulation and control of eddy current problems. Ph.D. thesis, Johannes Kepler University, Institute of Computational Mathematics, Linz, Austria (2012)Google Scholar
  11. 11.
    Kolmbauer, M.: Efficient solvers for multiharmonic eddy current optimal control problems with various constraints and their analysis. IMA J. Numer. Anal. (2012). doi: 10.1093/imanum/drs025zbMATHGoogle Scholar
  12. 12.
    Kolmbauer, M., Langer, U.: A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control problems. SIAM J. Sci. Comput. 34(6), B785–B809 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Paige, C.C., Saunders, M.A.: Solutions of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. AMS, Providence (2010)Google Scholar
  15. 15.
    Tröltzsch, F., Yousept, I.: PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages. ESAIM: M2AN 46, 709–729 (2012)Google Scholar
  16. 16.
    Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52(2), 559–581 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.DK Computational MathematicsJohannes Kepler University LinzLinzAustria
  2. 2.Institute of Computational MathematicsJohannes Kepler University LinzLinzAustria

Personalised recommendations