Skip to main content
SpringerLink
Log in
Book cover

Multiple Shooting and Time Domain Decomposition Methods pp 159–181Cite as

Direct Multiple Shooting for Parabolic PDE Constrained Optimization

  • Conference paper

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 9))

Abstract

Direct Multiple Shooting is a flexible and efficient method to solve difficult optimal control problems constrained by ordinary differential equations or differential-algebraic equations. The aim of this article is to concisely summarize the main conceptual and methodological approaches to solve also optimal control problems with parabolic partial differential equations constraints via a Direct Multiple Shooting method. The main obstacle is the sheer size of the discretized optimization problems. We explain a typical direct discretization approach and discuss an inexact SQP method based on two-grid Newton-Picard preconditioning. Special attention is given to a-posteriori κ-estimators that monitor contraction and to the structure-exploiting treatment of the resulting large-scale quadratic programming subproblems, including an extended condensing technique that exploits Multiple Shooting and two-grid Newton-Picard structures. Finally, we present numerical results for an advection-diffusion and a bacterial chemotaxis example.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Albersmeyer, J.: Adjoint based algorithms and numerical methods for sensitivity generation and optimization of large scale dynamic systems. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2010)

    Google Scholar 

  2. Albersmeyer, J., Bock, H.G.: Sensitivity generation in an adaptive BDF-method. In: Bock, H.G., Kostina, E., Phu, X.H., Rannacher, R. (eds.) Modeling, Simulation and Optimization of Complex Processes. Proceedings of the International Conference on High Performance Scientific Computing, pp. 15–24, Hanoi, 6–10 March 2006. Springer, Berlin (2008)

    Google Scholar 

  3. Bellman, R.E.: Dynamic Programming, 6th edn. University Press, Princeton (1957)

    MATH  Google Scholar 

  4. Biegler, L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)

    Article  Google Scholar 

  5. Bock, H.G.: Recent advances in parameter identification techniques for ODE. In: Deuflhard, P., Hairer, E. (eds.) Numerical Treatment of Inverse Problems in Differential and Integral Equations, pp. 95–121. Birkhäuser, Boston (1983)

    Chapter  Google Scholar 

  6. Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242–247, Budapest. Pergamon Press (1984)

    Google Scholar 

  7. Dautray, R., Lions, J.-L.: Evolution problems I. In: Craig, A. (ed.) Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1992)

    Google Scholar 

  8. Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2006)

    Google Scholar 

  9. Griewank, A., Walther, A.: Evaluating Derivatives, 2nd edn. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  10. Hesse, H.K.: Multiple shooting and mesh adaptation for PDE constrained optimization problems. Ph.D. thesis, University of Heidelberg (2008)

    Google Scholar 

  11. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, New York (2009)

    MATH  Google Scholar 

  12. Lebiedz, D., Brandt-Pollmann, U.: Manipulation of self-aggregation patterns and waves in a reaction-diffusion system by optimal boundary control strategies. Phys. Rev. Lett. 91(20), 208301 (2003)

    Article  Google Scholar 

  13. Lust, K., Roose, D., Spence, A., Champneys, A.R.: An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comput. 19(4), 1188–1209 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  15. Osborne, M.R.: On shooting methods for boundary value problems. J. Math. Anal. Appl. 27, 417–433 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pontryagin, L.S., Boltyanski, V.G., Gamkrelidze, R.V., Miscenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, Chichester (1962)

    MATH  Google Scholar 

  17. Potschka, A.: A direct method for the numerical solution of optimization problems with time-periodic PDE constraints. Ph.D. thesis, Universität Heidelberg (2011)

    Google Scholar 

  18. Potschka, A.: A Direct Method for Parabolic PDE Constrained Optimization Problems. Advances in Numerical Mathematics. Springer, Berlin (2013)

    Google Scholar 

  19. Potschka, A., Bock, H.G., Schlöder, J.P.: A minima tracking variant of semi-infinite programming for the treatment of path constraints within direct solution of optimal control problems. Optim. Methods Softw. 24(2), 237–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Potschka, A., Mommer, M.S., Schlöder, J.P., Bock, H.G.: Newton-Picard-based preconditioning for linear-quadratic optimization problems with time-periodic parabolic PDE constraints. SIAM J. Sci. Comput. 34(2), 1214–1239 (2012)

    Article  MathSciNet  Google Scholar 

  21. Russell, R.D., Shampine, L.F.: A collocation method for boundary value problems. Numer. Math. 19, 1–28 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelpha (2003)

    Book  MATH  Google Scholar 

  23. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  24. Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen, 2nd edn. Vieweg+Teubner Verlag, Wiesbaden (2009)

    Book  Google Scholar 

  25. Tsang, T.H., Himmelblau, D.M., Edgar, T.F.: Optimal control via collocation and non-linear programming. Int. J. Control. 21, 763–768 (1975)

    Article  MATH  Google Scholar 

  26. Tyson, R., Lubkin, S.R., Murray, J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Biol. 38, 359–375 (1999). doi:10.1007/s002850050153

    MathSciNet  MATH  Google Scholar 

  27. Tyson, R., Lubkin, S.R., Murray, J.D.: A minimal mechanism for bacterial pattern formation. Proc. R. Soc. B Biol. Sci. 266, 299–304 (1999)

    Article  Google Scholar 

  28. Walther, A., Kowarz, A., Griewank, A.: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. Technical report, Institute of Scientific Computing, Technical University Dresden (2005)

    Google Scholar 

  29. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). Translated from the German by C.B. Thomas and M.J. Thomas

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany

    Andreas Potschka

Authors
  1. Andreas Potschka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Potschka .

Editor information

Editors and Affiliations

  1. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Thomas Carraro

  2. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Michael Geiger

  3. Heidelberg University, Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

    Stefan Körkel

  4. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Rolf Rannacher

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Potschka, A. (2015). Direct Multiple Shooting for Parabolic PDE Constrained Optimization. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation