Skip to main content
SpringerLink
Log in

Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau–Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented to illustrate the effectiveness of the adaptivity for both regularization techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benedix, O., Vexler, B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. App. 44(1), 3–25 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bergounioux, M., Haddou, M., Hintermueller, M., Kunisch, K.: A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11, 495–521 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergounioux, M., Kunisch, K.: On the structure of lagrange multipliers for state-constrained optimal control prolems. Syst. Control Lett. 48, 169–176 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cangiani, A., Chapman, J., Georgoulis, E.H., Jensen, M.: On local super-penalization of interior penalty discontinuous Galerkin methods. Int. J. Numer. Anal. Mod. 11(3), 478–495 (2014)

  7. Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gong, W., Yan, N.: A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46, 182–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Günther, A., Hinze, M.: A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16, 307–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Günther, A., Hinze, M., Tber, M.H.: A Posteriori Error Representations for Elliptic Optimal Control Problems with Control and State Constraints. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol. 160, pp. 303–317. Springer, Basel (2012)

    Chapter  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Herzog, R., Sachs, E.: Preconditioned conjugate gradient methods for optimal control problems with control and state constraints. SIAM J. Matrix Anal. Appl. 31, 2291–2317 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hintermüller, M., Hinze, M.: Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47, 1666–1683 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hintermüller, M., Hoppe, R.H.W.: Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations. SIAM J. Control Optim. 48, 5468–5487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hintermüller, M., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM, Control Optim. Calc. Var. 14, 540–560 (2008)

  17. Hintermüller, M., Kunisch, K.: Pde-constrained optimization subject to pointwise constraints on the control, the state and its derivative. SIAM J. Optim. 20(3), 1133–1156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Comput. Optim. Appl. 46, 487–510 (2010)

  19. Hinze, M., Schiela, A.: Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment. Comput. Optim. Appl. 48, 581–600 (2011)

  20. Hinze, M., Yan, N., Zhou, Z.: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comp. Math. 27(2–3), 237–253 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Hoppe, R.H.W., Kieweg, M.: A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. J. Numer. Math. 17(3), 219–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hoppe, R.H.W., Kieweg, M.: Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput. Optim. Appl. 46(3), 511–533 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50, 221–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43(2), 213–233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Leykekhman, D.: Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems. J. Sci. Comput. 53, 483–511 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leykekhman, D., Heinkenschloss, M.: Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 50(4), 2012–2038 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41(5), 1321–1349 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22(6), 871–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation, Frontiers in Applied Mathematics, vol. 35. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)

    Book  Google Scholar 

  31. Rösch, A., Wachsmuth, D.: A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120, 733–762 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Schötzau, D., Zhu, L.: A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. 59(9), 2236–2255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15(2), 616–634 (2005)

    Article  MATH  Google Scholar 

  35. Verfürth, R.: Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wollner, W.: A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints. Comput. Optim. Appl. 47(1), 133–159 (2010)

  37. Yan, N., Zhou, Z.: A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math. 223(1), 198–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yücel, H., Heinkenschloss, M., Karasözen, B.: An adaptive discontinuous Galerkin method for convection dominated distributed optimal control problems. Technical Report, pp. 77005–1892. Department of Computational and Applied Mathematics, Rice University, Houston (2012)

  39. Yücel, H., Heinkenschloss, M., Karasözen, B.: Distributed optimal control of diffusion-convection-reaction equations using discontinuous Galerkin methods. In: Numerical Mathematics and Advanced Applications 2011, pp. 389–397. Springer, Berlin, Heidelberg (2013)

  40. Yücel, H., Karasözen, B.: Adaptive symmetric interior penalty galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints. Optimization 63, 145–166 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou, Z., Yu, X., Yan, N.: Local discontinuous Galerkin approximation of convection-dominated diffusion optimal control problems with control constraints. Numer. Methods Partial Differ. Equ. 30(1), 339–360 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5(1), 49–62 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Martin Stoll for helpful discussions about this research. This work was supported by the Forschungszentrum Dynamische Systeme (CDS): Biosystemtechnik, Otto-von-Guericke-Universität Magdeburg. The authors also would like to express their sincere thanks to the referees for most valuable suggestions.

Author information

Authors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Hamdullah Yücel & Peter Benner

Authors
  1. Hamdullah Yücel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Benner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamdullah Yücel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yücel, H., Benner, P. Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations. Comput Optim Appl 62, 291–321 (2015). https://doi.org/10.1007/s10589-014-9691-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-014-9691-7

Keywords

Search

Navigation