[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

Generalized Multilevel SQP-methods for PDAE-constrained Optimization Based on Space-Time Adaptive PDAE Solvers

  • Chapter
  • First Online:
Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

Abstract

In this work, we present an all-in-one optimization approach suitable to solve complex optimal control problems with time-dependent nonlinear partial differential algebraic equations and point-wise control constraints. A newly developed generalized SQP-method is combined with an error based multilevel strategy and the state-of-the-art software package Kardos to allow the efficient resolution of different space and time scales in an adaptive manner. The numerical performance of the method is demonstrated and analyzed for a real-life two-dimensional radiative heat transfer problem modelling the optimal boundary control for a cooling process in glass manufacturing.

Mathematics Subject Classification (2000). 49J20, 49M25, 65C20, 65K10, 65M60, 90C46, 90C55.

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

Access this chapter

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 103.50
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 129.99
Price includes VAT (United Kingdom)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
GBP 129.99
Price includes VAT (United Kingdom)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. R. Becker, M. Braack, D. Meidner, R. Rannacher, and B. Vexler. Adaptive finite element methods for PDE-constrained optimal control problems. In Reactive flows, diffusion and transport, pages 177–205. Springer, Berlin, 2007.

    Chapter  Google Scholar 

  2. A. Borzì. Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci., 11(1):59–66, 2008.

    Article  MathSciNet  Google Scholar 

  3. A. Borzì and K. Kunisch. A multigrid scheme for elliptic constrained optimal control problems. Comput. Optim. Appl., 31(3):309–333, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.H. Byrd. Robust trust region methods for constrained optimization. In Third SIAM Conference on Optimization, Houston, TX, 1987.

    Google Scholar 

  5. D. Clever and J. Lang. Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient. Preprint 2595, Technische Universität Darmstadt, 2009.

    Google Scholar 

  6. D. Clever, J. Lang, S. Ulbrich, and J.C. Ziems. Combination of an adaptive multilevel SQP method and a space-time adaptive PDAE solver for optimal control problems. Appears in Procedia Computer Science, 2010.

    Google Scholar 

  7. Andrew R. Conn, N.I.M. Gould, and P.L. Toint. Trust-region methods. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.

    Book  Google Scholar 

  8. K. Debrabant and J. Lang. On global error estimation and control for parabolic equations. Technical Report 2512, Technische Universität Darmstadt, 2007.

    Google Scholar 

  9. J.E. Dennis, Jr., M. El-Alem, and M.C. Maciel. A global convergence theory for general trust-region-based algorithms for equality constrained optimization. SIAM J. Optim., 7(1):177–207, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Deuflhard. Global inexact Newton methods for very large scale nonlinear problems. In IMPACT of Computing in Science and Engineering, pages 366–393, 1991.

    Google Scholar 

  11. P. Deuflhard, P. Leinen, and H. Yserentant. Concepts of an adaptive hierarchical finite element code. Impact of Comput. in Sci. and Engrg., 1:3–35, 1989.

    Article  MATH  Google Scholar 

  12. T. Dreyer, B. Maar, and V. Schulz. Multigrid optimization in applications. J. Comput. Appl. Math., 120(1-2):67–84, 2000. SQP-based direct discretization methods for practical optimal control problems.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Erdmann, J. Lang, and R. Roitzsch. KARDOS-User’s Guide. Manual, Konrad- Zuse-Zentrum Berlin, 2002.

    Google Scholar 

  14. S. Gratton, M. Mouffe, Ph.L. Toint, and M. Weber-Mendonca. A recursive trustregion method in infinity norm for bound-constrained nonlinear optimization. IMA Journal of Numerical Analysis, 2008 (to appear).

    Google Scholar 

  15. S. Gratton, A. Sartenaer, and P.L. Toint. Recursive trust-region methods for multiscale nonlinear optimization. SIAM J. Optim., 19(1):414–444, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, Second Revised Edition. Springer Verlag, 2002.

    Google Scholar 

  17. M. Heinkenschloss and L.N. Vicente. Analysis of inexact trust-region SQP algorithms. SIAM J. Optim., 12(2):283–302, 2001/02.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints. Mathematical Modelling: Theory and Applications (23). Springer, 2008.

    Google Scholar 

  19. C.T. Kelley. Iterative Methods for Optimization. Frontiers in Applied Mathematics. SIAM, 1999.

    Google Scholar 

  20. J. Lang. Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications, volume 16 of Lecture Notes in Computational Science and Engineering. Springer Verlag, 2000.

    Google Scholar 

  21. J. Lang. Adaptive computation for boundary control of radiative heat transfer in glass. Journal of Computational and Applied Mathematics, 183:312–326, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Lang and D. Teleaga. Towards a fully space-time adaptive FEM for magnetoquasistatics. IEEE Transactions on Magnetics, 44,6:1238–124, 2008.

    Article  Google Scholar 

  23. J. Lang and J. Verwer. ROS3P – an accurate third-order Rosenbrock solver designed for parabolic problems. BIT, 41:730–737, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Lang and J. Verwer. On global error estimation and control for initial value problems. SIAM J. Sci. Comput., 29:1460–1475, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  25. E.W. Larsen, G. Thömmes, A. Klar, M. Seaïd, and T. Götz. Simplified P𝑁 approximations to the equations of radiative heat transfer and applications. Journal of Computational Physics, 183:652–675, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  26. R.M. Lewis and S.G. Nash. Model problems for the multigrid optimization of systems governed by differential equations. SIAM J. Sci. Comput., 26(6):1811–1837 (electronic), 2005.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Meidner and B. Vexler. Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim., 46(1):116–142 (electronic), 2007.

    Article  MathSciNet  MATH  Google Scholar 

  28. E.O. Omojokun. Trust region algorithms for optimization with nonlinear equality and inequality constraints. PhD thesis, University of Colorado, Boulder, Colorado, USA, 1989.

    Google Scholar 

  29. L. Petzold. Adjoint sensitivity analysis for time-dependent partial differential equations with adaptive mesh refinement. Journal of Computational Physics (198), 310-325, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Steihaug. The conjugate gradient method and trust regions in large scale optimization.SIAM, Journal of Numerical Analysis, 20,1:626–637, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Tröltzsch. Optimale Steuerung partieller Differentialgleichungen. Vieweg, 2005.

    Google Scholar 

  32. S. Ulbrich. Generalized SQP methods with “parareal” time-domain decomposition for time-dependent PDE-constrained optimization. In Real-time PDE-constrained optimization, volume 3 of Comput. Sci. Eng., pages 145–168. SIAM, Philadelphia, PA, 2007.

    Chapter  Google Scholar 

  33. M. Vallejos and A. Borzì. Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing, 82(1):31–52, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  34. H.A. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 13:631–644, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Verfürth. A posteriori error estimators for convection-diffusion equations. Numer. Math., 80(4):641–663, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Verfürth. A posteriori error estimates for linear parabolic equations. Preprint, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, Germany, 2004.

    Google Scholar 

  37. R. Verfürth. A posteriori error estimates for non-linear parabolic equations. Preprint, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, Germany, 2004.

    Google Scholar 

  38. J.C. Ziems. Adaptive Multilevel SQP-methods for PDE-constrained optimization. Dissertation, Fachbereich Mathematik, Technische Universität Darmstadt. Verlag Dr. Hut, München, 2010.

    Google Scholar 

  39. J.C. Ziems and S. Ulbrich. Adaptive multilevel inexact SQP methods for PDEconstrained optimization. SIAM J. Optim., 21(1):1–40, 2011.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Debora Clever .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Clever, D., Lang, J., Ulbrich, S., Ziems, C. (2012). Generalized Multilevel SQP-methods for PDAE-constrained Optimization Based on Space-Time Adaptive PDAE Solvers. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_4

Download citation

Publish with us

Policies and ethics