Advances in Computational Mathematics

, Volume 41, Issue 2, pp 457–488 | Cite as

Second-order approximation and fast multigrid solution of parabolic bilinear optimization problems

  • Alfio BorzìEmail author
  • Sergio González Andrade


An accurate and fast solution scheme for parabolic bilinear optimization problems is presented. Parabolic models where the control plays the role of a reaction coefficient and the objective is to track a desired trajectory are formulated and investigated. Existence and uniqueness of optimal solution are proved. A space-time discretization is proposed and second-order accuracy for the optimal solution is discussed. The resulting optimality system is solved with a nonlinear multigrid strategy that uses a local semismooth Newton step as smoothing scheme. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the ability of the multigrid scheme to solve the given optimization problems with mesh-independent efficiency.


Multigrid methods Newton methods Finite differences Parabolic partial differential equations Bilinear control ptimal control theory 


49K20 49J20 65M55 65C20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Addou, A. , Benbrik, A.: Existence and Uniqueness of Optimal Control for a Distributed-Parameter Bilinear System. J. Dyn. Control Syst. 8, 141–152 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Borzì, A., González Andrade, S.: Multigrid solution of a linear Lavrentiev-regularized state-constrained parabolic control problem, Numerical Mathematics. Theory Methods Appl. 5, 1–18 (2012)zbMATHGoogle Scholar
  3. 3.
    Borzì, A., Kunisch, K.: A multigrid scheme for elliptic constrained optimal control problems. Comput. Optim. Appl. 31, 309–333 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations,SIAM,Philadelphia,2012.Google Scholar
  5. 5.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31, 333–390 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brzezniak, Z.: On smooth dependence of solutions od parabolic equations on coefficients. Univ. Iagellonicae Acta Math. 29, 7–17 (1992)MathSciNetGoogle Scholar
  7. 7.
    Emmrich, E.: Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations. Comput. Methods Appl. Math. 9, 37–62 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence (2002)Google Scholar
  9. 9.
    González Andrade, S., Borzì, A.: Multigrid second-order accurate solution of parabolic control-constrained problems. Comput. Optim. Appl. 51, 835–866 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods ,Wiley, (1995)Google Scholar
  11. 11.
    Hackbusch, W.: Elliptic Differential Equations. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hager, W.W., Zhang, H.: A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search. SIAM J. Optim. 16, 170–192 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hintermüller, M, Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Opt. 13, 865–888 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kärkkäinen, T.: Error estimates for distributed parameter identification in linear elliptic equations. J. Math. Sys. Estimation Control 6, 1–20 (1996)Google Scholar
  15. 15.
    Kärkkäinen, T.: Error estimates for distributed parameter identification in parabolic problems with output least squares and Crank-Nicolson method. Appl. Math. 42, 259–277 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kröner, A., Vexler, B.: A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Applied Math. 230, 781—802 (2009)CrossRefGoogle Scholar
  17. 17.
    Kunisch, K., Tai, X.-C.: Sequential and Parallel Splitting Methods for Bilinear Control Problems in Hilbert Spaces, SIAM. J. Numer. Anal. 34, 91–118 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ladyzhenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type, American Mathematical Society Translations. RI, Providence (1968)Google Scholar
  19. 19.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  20. 20.
    Meserve, B.E.: Fundamental Concepts of Algebra. Dover Publications, USA (1982)Google Scholar
  21. 21.
    Mingione, G.: Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Appl. Math. 51, 355–426 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Nash, J.: Continuity of solutions of parabolic and elliptic Equations. Am. J. Math. 80, pp.931–954 (1958)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nittka, R. University of Ulm, PhD Thesis (2010)Google Scholar
  24. 24.
    Press, W., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, USA (1997)Google Scholar
  25. 25.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Springer, USA (2000)Google Scholar
  26. 26.
    Stadler, G: Semismooth Newton and augmented Lagrangian methods for a simplified friction problem, SIAM. J. Optim. 15, 39—62 (2004)MathSciNetGoogle Scholar
  27. 27.
    Tagiev, R.K, Optimal coefficient control in parabolic systems. Diff. Equat. 45, 1526–1535 (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Tai, X.-C., Neittaanmäki, P.: Error estimates for numerical identification of distributed parameters. J. Comp. Math. 10, 66–78 (1992)zbMATHGoogle Scholar
  29. 29.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. AMS, USA (2010)CrossRefGoogle Scholar
  30. 30.
    Trottenberg, U., Oosterlee, C., and Schüller, A.: Multigrid. Academic Press, London (2001)zbMATHGoogle Scholar
  31. 31.
    Vallejos, M., and Borzì, A.: Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing 82, pp. 31–52 (2008)CrossRefGoogle Scholar
  32. 32.
    Wloka, J. Cambridge University Press, Partial Differential Equations (1987)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Research Center on Mathematical Modelling MODEMATEscuela Politécnica NacionalQuitoEcuador

Personalised recommendations