Skip to main content
SpringerLink
Log in

Multigrid second-order accurate solution of parabolic control-constrained problems

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

Abstract

A mesh-independent and second-order accurate multigrid strategy to solve control-constrained parabolic optimal control problems is presented. The resulting algorithms appear to be robust with respect to change of values of the control parameters and have the ability to accommodate constraints on the control also in the limit case of bang-bang control. Central to the development of these multigrid schemes is the design of iterative smoothers which can be formulated as local semismooth Newton methods. The design of distributed controls is considered to drive nonlinear parabolic models to follow optimally a given trajectory or attain a final configuration. In both cases, results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed schemes are able to solve parabolic optimality systems with textbook multigrid efficiency. Further results are presented to validate second-order accuracy and the possibility to track a trajectory over long time intervals by means of a receding-horizon approach.

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.

Similar content being viewed by others

References

  1. Allgöwer, F., Badgwell, T.A., Qin, J.S., Rawlings, J.B., Wright, S.J.: Nonlinear predictive control and moving horizon estimation—An introduction overview. In: Frank, P.M. (ed.) Advances in Control, Highlights of ECC’99, pp. 391–449 (1999), Chap. 5

    Google Scholar 

  2. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)

    Book  MATH  Google Scholar 

  3. Batista, M.: A method for solving cyclic block penta-diagonal systems of linear equations (2008). arXiv:0803.0874v3

  4. Batista, M., Karawia, A.A.: The use of the Sherman-Morrison-Woodbury formula to solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations. Appl. Math. Comput. 210, 558–563 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benkert, K., Fischer, R.: An efficient implementation of the Thomas-Algorithm for block penta-diagonal systems on vector computers. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) Proceedings of the 7th International Conference on Computer Science, ICCS (2007)

    Google Scholar 

  6. Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) Real-Time PDE-Constrained Optimization, Computational Science and Engineering, vol. 3. SIAM, Philadelphia (2007)

    Google Scholar 

  7. Borzì, A.: High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems. J. Comput. Appl. Math. 200, 67–85 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borzì, A.: Multigrid methods for parabolic distributed optimal control problems. J. Comput. Appl. Math. 157, 365–382 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borzì, A.: Space-time multigrid methods for solving unsteady optimal control problems. In: Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) Real-Time PDE-Constrained Optimization, Computational Science and Engineering, vol. 3. SIAM, Philadelphia (2007), Chap. 5

    Google Scholar 

  10. Borzì, A., Griesse, R.: Distributed optimal control of lambda-omega systems. J. Numer. Math. 14, 17–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borzì, A., Kunisch, K.: A multigrid scheme for elliptic constrained optimal control problems. Comput. Optim. Appl. 31, 309–333 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Borzì, A., von Winckel, G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. 31, 2172–2192 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brézis, H., Crandall, M.-G., Pazy, A.: Perturbations of nonlinear maximal monotone sets in Banach spaces. Commun. Pure Appl. Math. 23, 123–144 (1970)

    Article  MATH  Google Scholar 

  16. Dreyer, Th., Maar, B., Schulz, V.: Multigrid optimization in applications. J. Comput. Appl. Math. 120, 67–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    MathSciNet  MATH  Google Scholar 

  18. Glashoff, K., Sachs, E.: On theoretical and numerical aspects of the bang-bang principle. Numer. Math. 29, 93–113 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goldberg, H., Tröltzsch, F.: Second order sufficient optimality conditions for a class of non-linear parabolic boundary control problems. SIAM J. Control Optim. 31, 1007–1027 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goldberg, H., Tröltzsch, F.: On a SQP–multigrid technique for nonlinear parabolic boundary control problems. In: Hager, W.W., Pardalos, P.M. (eds.) Optimal Control: Theory, Algorithms, and Applications, pp. 154–174. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  21. Hackbusch, W.: Parabolic multigrid methods. In: Glowinski, R., Lions, J.-L. (eds.) Computing Methods in Applied Sciences and Engineering VI. North-Holland, Amsterdam (1984)

    Google Scholar 

  22. Hackbusch, W.: Elliptic Differential Equations. Springer, New York (1992)

    MATH  Google Scholar 

  23. Hackbusch, W.: On the fast solving of parabolic boundary control problems. SIAM J. Control Optim. 17, 231–244 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hackbusch, W.: Numerical Solution of Linear and Nonlinear Parabolic Optimal Control Problems. Lecture Notes in Control and Information Science, vol. 30. Springer, Berlin (1981)

    Google Scholar 

  25. Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2003)

    Article  MATH  Google Scholar 

  26. Hintermüller, M., Hoppe, R.H.W.: Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47, 1721–1743 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16(4), 848–864 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ito, K., Kunisch, K.: Asymptotic properties of receding horizon optimal control problems. SIAM J. Control Optim. 40(5), 1585–1610 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. ESAIM: Math. Model. Numer. Anal. 42, 1–23 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  32. Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim. 8, 69–95 (1981)

    Article  MathSciNet  Google Scholar 

  33. Meidner, D., Vexler, B.: Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46, 116–142 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nagaiah, Ch., Kunisch, K., Plank, G.: Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. Comput. Optim. Appl. doi:10.1007/s10589-009-9280-3

  35. Neittaanmäki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems. Dekker, New York (1994)

    MATH  Google Scholar 

  36. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–368 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw. 21, 121–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs AMS. AMS, Providence (1997)

    MATH  Google Scholar 

  39. Stadler, G.: Semismooth Newton and augmented Lagrangian methods for a simplified friction problem. SIAM J. Optim. 15, 39–62 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, Berlin (1995)

    MATH  Google Scholar 

  41. Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, London (2001)

    MATH  Google Scholar 

  42. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–842 (2002)

    Article  MathSciNet  Google Scholar 

  43. Varga, S.R.: Matrix Iterative Analysis. Prentice Hall, New York (1962)

    Google Scholar 

  44. Zuazua, E.: Switching control. J. Eur. Math. Soc. (to appear)

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010, Graz, Austria

    S. González Andrade & A. Borzì

  2. Research Group on Optimization, Departamento de Matemática, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, Quito, Ecuador

    S. González Andrade

  3. Dipartimento e Facoltà di Ingegneria, Palazzo Dell’Aquila Bosco Lucarelli, Università degli Studi del Sannio, Corso Garibaldi 107, 82100, Benevento, Italia

    A. Borzì

Authors
  1. S. González Andrade
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Borzì
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Borzì.

Additional information

Supported in part by the Austrian Science Fund SFB Project F3205-N18 “Fast Multigrid Methods for Inverse Problems”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

González Andrade, S., Borzì, A. Multigrid second-order accurate solution of parabolic control-constrained problems. Comput Optim Appl 51, 835–866 (2012). https://doi.org/10.1007/s10589-010-9358-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-010-9358-y

Keywords

Search

Navigation