Skip to main content
SpringerLink
Log in
Book cover

Trends in PDE Constrained Optimization pp 251–284Cite as

Birkhäuser

Model Reduction by Adaptive Discretization in Optimal Control

  • Chapter
  • First Online:

  • 1775 Accesses

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

Abstract

This article gives a survey of recent developments in the economical numerical solution of PDE-based optimal control problems by adaptive methods. Systematic mesh adaptivity combined with adaptive stopping criteria for linear and nonlinear algebraic iterations is one approach to reducing the computational cost to an acceptable level. These various steps of adaptivity are driven by “goal-oriented” a posteriori error estimates derived within the general framework of the DWR (Dual Weighted Residual) method. The presented material is mainly based on results obtained within the second funding period 2011–2013 of this subproject of the DFG Priority Program 1253 “Optimization with Partial Differential Equations”. In this sense it is the continuation of the article “A posteriori error estimation in PDE-constrained optimization with pointwise inequality constraints” by R. Rannacher, B. Vexler, and W. Wollner in “Constrained Optimization and Optimal Control for Partial Differential Equations” (G. Leugering et al., eds), Birkhüser, Basel, 2012.

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   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   139.99
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. M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis (Wiley-Interscience, New York, 2000)

    Book  MATH  Google Scholar 

  2. I. Babuška, Th. Strouboulis, The Finite Element Method and Its Reliability (The Clarendon Press/Oxford University Press, New York, 2001)

    Google Scholar 

  3. G. Bal, On the convergence and stability of the parareal algorithm to solve partial differential equations, in Proceedings of the 15th International Domain Decomposition Conference. Lecture Notes in Computer Science and Engineering (Springer, Berlin, 2003), pp. 426–432

    Google Scholar 

  4. W. Bangerth, R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2003)

    Google Scholar 

  5. R. Becker, Mesh adaptation for stationary flow control. J. Math. Fluid Mech. 3, 317–341 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Becker, Estimating the control error in discretized PDE-constraint optimization. J. Numer. Math. 14, 163–185 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Becker, H. Kapp, R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39, 113–132 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Becker, D. Meidner, B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22, 813–833 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Becker, R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)

    MATH  MathSciNet  Google Scholar 

  10. R. Becker, R. Rannacher, Weighted a-posteriori error estimates in FE methods, in Lecture ENUMATH-95, Paris, 1995; Proceedings of ENUMATH-97, Heidelberg, 1997, ed. by H.G. Bock et al. (World Science Publisher, Singapore, 1998) pp. 621–637

    Google Scholar 

  11. R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation, in Acta Numerica 2001, ed. by A. Iserles (Cambridge University Press, Cambridge, 2001), pp. 1–102

    Google Scholar 

  12. R. Becker, B. Vexler, A posteriori error estimation for finite element discretization of parameter identification problems. Numer. Math. 96, 435–459 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Becker, B. Vexler, Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations. J. Comput. Phys. 206, 95–110 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Bergounioux, A penalization method for optimal control of elliptic problems with state constraints. SIAM J. Control Optim. 30, 305–323 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Besier, R. Rannacher, Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Meth. Fluids 2011. Published online doi:10.1002/fld.2735

    Google Scholar 

  18. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 2nd edn. (Springer, Berlin, 2002)

    Google Scholar 

  19. G.F. Carey, J.T. Oden, Finite Elements. Computational Aspects (Prentice-Hall, Englewood Cliffs, 1984)

    Google Scholar 

  20. T. Carraro, M. Geiger, R. Rannacher, Indirect multiple shooting for nonlinear parabolic optimal control problems with control constraints. SIAM J. Sci. Comput. 36, A452–A481 (2014). Online publication http://epubs.siam.org/toc/sjoce3/36/2. doi:10.1137/120895809

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

    Article  MATH  MathSciNet  Google Scholar 

  22. E. Casas, L.A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim. 27, 35–56 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. R. Dautray, J.-L. Lions, Evolution Problems I. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5 (Springer, Berlin, 1992)

    Google Scholar 

  24. K. Deckelnick, A. Günther, M. Hinze, Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111, 335–350 (2008)

    Article  Google Scholar 

  25. F. de Hoog, R.M.M. Mattheij, On dichotomy and well conditioning in BVP. SIAM J. Num. Anal. 24, 89–105 (1987)

    Article  MATH  Google Scholar 

  26. P. Deuflhard, A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22, 289–315 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  27. J.C. Dunn, Global and asymptotic convergence rate estimates for a class of projected gradient processes. SIAM J. Control Optim. 19, 368–400 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  28. K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Computational Differential Equations (Cambridge University Press, Cambridge, 1996)

    MATH  Google Scholar 

  29. A. Gaevskaya, R.H.W. Hoppe, Y. Iliash, M. Kieweg, Convergence analysis of an adaptive finite element method for distributed control problems with control constraints, in Control of Coupled Partial Differential Equations, ed. by G. Leugering et al. (Birkhäuser, Basel, 2007)

    Google Scholar 

  30. M. Gander, S. Vandewalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556–578 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Griesbaum, B. Kaltenbacher, B. Vexler, Efficient computation of the Tikhonov regularization parameter by goal-oriented adaptive discretization. Inverse Probl. 24(2), 025025 (2008)

    Google Scholar 

  32. R. Griesse, B. Vexler, Numerical sensitivity analysis for the quantity of interest in pde-constrained optimization. SIAM J. Sci. Comput. 29, 22–48 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. R. Griesse, S. Volkwein, A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system. SIAM J. Control Optim. 44, 467–494 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Günther, M. Hinze, M.H. Tber, A posteriori error representations for elliptic optimal control problems with control and state constraints, in Constrained Optimization and Optimal Control for Partial Differential Equations, ed. by G. Leugering et al. (Springer, Basel, 2012), pp. 303–317

    Chapter  Google Scholar 

  36. M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. J. Comput. Appl. Math. 173, 169–198 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. R. Herzog, K. Kunisch, Algorithms for pde-constrained optimization. GAMM Rep. 33, 163–176 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. H.K. Hesse, Multiple shooting and mesh adaptation for PDE constrained optimization problems, doctoral dissertation, Heidelberg University, 2008

    Google Scholar 

  39. H.K. Hesse, G. Kanschat, Mesh adaptive multiple shooting for partial differential equations. Part 1: linear-quadratic optimal control problems. J. Numer. Math. 17, 195–217 (2009)

    Google Scholar 

  40. N.T. Hieu, Remarks on the shooting method for nonlinear two-point boundary-value problems. VNU J. Sci. 3, 18–25 (2003)

    Google Scholar 

  41. M. Hintermüller, M. Hinze, Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47, 1666–1683 (2008)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  45. M. Hintermüller, K. Kunisch, Stationary optimal control problems with pointwise state constraints. SIAM J. Optim. 20, 1133–1156 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  47. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23 (Springer, New York, 2009)

    Google Scholar 

  48. M. Hinze, A. Schiela, Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment. Comput. Optim. Appl. (2009) doi:10.1007/s10589-009-9278-x

    Google Scholar 

  49. R.H.W. Hoppe, Y. Iliash, Ch. Iyyunni, N.H. Sweilam, A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Math. 14, 57–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  51. T. Jankowski, Approximate solutions of boundary value problems for systems of ordinary differential equations. Zh. Vychisl. Mat. Mat. Fiz. 35, 1050–1057 (1995)

    MathSciNet  Google Scholar 

  52. B. Kaltenbacher, A. Kirchner, B Vexler, Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems. Inverse Probl. 27, 125008 (2011)

    MathSciNet  Google Scholar 

  53. H. Kapp, Adaptive Finite elements for optimization in partial differential equations, doctoral dissertation, Heidelberg University, 2000

    Google Scholar 

  54. C.T. Kelley, E.W. Sachs, Solution of optimal control problems by a pointwise projected Newton method. SIAM J. Control Optim. 33, 1731–1757 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  55. S. Körkel, E. Kostina, H.G. Bock, J.P. Schlöder, Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optim. Methods Softw. 19, 327–338 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  56. K. Kunisch, A. Rösch, Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13, 321–334 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  57. D.B. Leineweber, I. Bauer, H.G. Bock, and J.P. Schlöder, An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part I: theoretical aspects. Comput. Chem. Eng. 27, 157–166 (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  59. J.-L. Lions, Y. Maday, G. Turinici, Resolution d’edp par un schema en temps ‘parareel’. C. R. Acad. Sci. Paris Ser. I 332, 661–668 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  60. W. Liu, Adaptive multi-meshes in finite element approximation of optimal control. Contemp. Math. 383, 113–132 (2005)

    Article  Google Scholar 

  61. W. Liu, N. Yan, A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15, 285–309 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  62. Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations. C. R. Acad. Sci. Paris Ser. I 335, 387–392 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  63. M.R. Maier, An adaptive shooting method for singularly perturbed boundary value problems. SIAM J. Sci. Stat. Comput. 7, 418–440 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  64. R.M.M. Mattheij, G.W.M. Staarink, On optimal shooting intervals. Math. Comput. 42, 25–40 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  65. D. Meidner, Adaptive space-time finite element methods for optimiza- tion problems governed by nonlinear parabolic systems, doctoral dissertation, Heidelberg University, 2008.

    Google Scholar 

  66. D. Meidner, R. Rannacher, B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. ControlOptim. 49, 1961–1997 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  67. D. Meidner, R. Rannacher, J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations. J. Numer. Math. 17, 143–172 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  69. D. Meidner, B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47, 1150–1177 (2008); Part II: problems with control constraints. SIAM J. Control Optim. 47, 1301–1329 (2008)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  71. C. Ortner, W. Wollner, A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state. Numer. Math. 118, 587–600 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  72. R. Rannacher, B. Vexler, W. Wollner, A posteriori error estimation in PDE-constrained optimization with pointwise inequality constraint, in Constrained Optimization and Optimal Control for Partial Differential Equations, ed. by G. Leugering et al. (Birkhüser, Basel, 2012), pp. 349–373

    Chapter  Google Scholar 

  73. R. Rannacher, J. Vihharev, Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors. J. Numer. Math. 21, 23–62 (2010)

    MathSciNet  Google Scholar 

  74. R. Rannacher, A. Westenberger, W. Wollner, Adaptive finite element approximation of eigenvalue problems: balancing discretization and iteration error. J. Numer. Math. 18, 303–327 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  75. A. Schiela, W. Wollner, Barrier methods for optimal control problems with convex nonlinear gradient state constraints. SIAM J. Optim. 21, 269–286 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  76. M. Schmich, Adaptive finite element methods for computing nonstationary incompressible flow, doctoral dissertation, Heidelberg University, 2009

    Google Scholar 

  77. M. Schmich, B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30, 369–393 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  78. R. Serban, S. Li, L. Petzold, Adaptive algorithms for optimal control of time-dependent partial differential-algebraic systems. Int. J. Numer. Methods Eng. 57, 1457–1469 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  79. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (American Mathematical Society, Providence, 2010)

    Google Scholar 

  80. S. Ulbrich, generalized SQP-methods with ‘parareal’ time-domain decomposition for time-dependent PDE-constrained optimization, in Real-time PDE-constrained optimization (SIAM, Philadelphia, 2007), pp. 145–168

    Google Scholar 

  81. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley/Teubner, New York/Stuttgart, 1996)

    MATH  Google Scholar 

  82. B. Vexler, W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47, 509–534 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  83. R. Weiss, The convergence of shooting methods. BIT 13, 470–475 (1973)

    Article  MATH  Google Scholar 

  84. W. Wollner, Adaptive FEM for PDE constrained optimization with point-wise constraints on the gradient of the state. Proc. Appl. Math. Mech. 8, 10873 (2009)

    Article  Google Scholar 

  85. W. Wollner, Adaptive methods for PDE-based optimal control with pointwise inequality constraints, doctoral dissertation, Heidelberg University, 2010

    Google Scholar 

  86. W. Wollner, 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, 133–159 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  87. W. Wollner, Goal-oriented adaptivity for optimization of elliptic systems subject to pointwise inequality constraints: application to free material optimization. Proc. Appl. Math. Mech. 10, 669–672 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported during the period 2007–2013 within the DFG Priority Program 1253 “Optimization with Partial Differential Equations”, grant 306/15-1, “Model reduction by adaptive discretization in optimal control”.

Author information

Authors and Affiliations

  1. Institut für angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, 69120, Heidelberg, Germany

    Rolf Rannacher

Authors
  1. Rolf Rannacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rolf Rannacher .

Editor information

Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Rannacher, R. (2014). Model Reduction by Adaptive Discretization in Optimal Control. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_17

Download citation

Publish with us

Policies and ethics

Search

Navigation