Numerical Algorithms

, Volume 73, Issue 3, pp 631–663 | Cite as

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Poisson and convection-diffusion control
Original Paper

Abstract

Saddle point matrices of a special structure arise in optimal control problems. In this paper we consider distributed optimal control for various types of scalar stationary partial differential equations and compare the efficiency of several numerical solution methods. We test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain assumptions the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and elapsed time is favourably compared with other published methods.

Keywords

PDE-constrained optimization problems Finite elements Iterative solution methods Preconditioning 

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References

  1. 1.
    Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. 65, 301–314 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Axelsson, O., Neytcheva, M., Bashir Ahmad, A: Comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Alg. 66, 811–841 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bangerth, W., Hartmann, R., Kanschat, G.: deal.II-a general-purpose object-oriented finite element library. ACM T. Math. Software 33 (2007). doi:10.1145/1268776.1268779. Art. 24
  6. 6.
    Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boyanova, P., Do-Quang, M., Neytcheva, M.: Efficient preconditioners for large scale binary Cahn-Hilliard models. Comput. Methods Appl. Math. 12, 1–22 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Choi, Y.: Simultaneous analysis and design in PDE-constrained optimization. Doctor of Philosophy Thesis, Stanford University (2012)Google Scholar
  9. 9.
    Greenbaum, A., Ptak, V., Strakos, Z.: Any convergence curve is possible for GMRES. SIAM Matrix Anal. Appl. 17, 465–470 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Heinkenschloss, M., Leykekhman, D.: Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47, 4607–4638 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constrains. Springer, Berlin Heidelberg New York (2009)MATHGoogle Scholar
  12. 12.
    Kollmann, M.: Efficient iterative solvers for saddle point systems arising in PDE-constrained optimization problems with inequality constraints. Doctor of Philosophy Thesis, Johannes Kepler University Linz (2013)Google Scholar
  13. 13.
    Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  14. 14.
    Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Alg. Appl. 19, 816–829 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pearson, J.W., Wathen, A.J.: Fast iterative solvers for convection-diffusion control problems. ETNA 40, 294–310 (2013)MathSciNetMATHGoogle Scholar
  16. 16.
    Rees, T.: Preconditioning iterative methods for PDE constrained optimization. Doctor of Philosophy Thesis, University of Oxford (2010)Google Scholar
  17. 17.
    The Trilinos Project http://trilinos.sandia.gov/
  18. 18.
    Tröltzsch, F.: Optimal control of partial differential equations: theory, methods and applications, AMS, Graduate Studies in Mathematics (2010)Google Scholar
  19. 19.
    Zulehner, W.: Nonstandard norms and robust estimates for saddle-point problems. SIAM J. Matrix Anal. Appl. 32, 536–560 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Zulehner, W.: Efficient solvers for saddle point problems with applications to PDE-constrained optimization Advanced Finite Element Methods and Applications, Lecture Notes in Applied and Computational Mechanics, vol. 66, pp 197–216 (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Geonics AS CROstravaCzech Republic
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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