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A Posteriori Error Estimation in PDE-constrained Optimization with Pointwise Inequality Constraints

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Constrained Optimization and Optimal Control for Partial Differential Equations

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

Abstract

This article summarizes several recent results on goal-oriented error estimation and mesh adaptation for the solution of elliptic PDE-constrained optimization problems with additional inequality constraints. The first part is devoted to the control constrained case. Then some emphasis is given to pointwise inequality constraints on the state variable and on its gradient. In the last part of the article regularization techniques for state constraints are considered and the question is addressed, how the regularization parameter can adaptively be linked to the discretization error.

Mathematics Subject Classification (2000). 65N30, 65K10; 90C59.

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References

  1. Mark Ainsworth and John Tinsley Oden. A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics (New York). Wiley- Interscience [John Wiley & Sons], New York, 2000.

    MATH  Google Scholar 

  2. Roland Becker. Estimating the control error in discretized PDE-constraint optimization. J. Numer. Math., 14(3):163–185, 2006.

    MathSciNet  MATH  Google Scholar 

  3. Maïtine Bergounioux. A penalization method for optimal control of elliptic problems with state constraints. SIAM J. Control Optim., 30(2):305–323, 1992.

    MathSciNet  MATH  Google Scholar 

  4. Maïtine Bergounioux. On boundary state constrained control problems. Numer. Funct. Anal. Optim., 14(5&6):515–543, 1993.

    MathSciNet  MATH  Google Scholar 

  5. Roland Becker, Hartmut Kapp, and Rolf Rannacher. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim., 39(1):113–132, 2000.

    MathSciNet  MATH  Google Scholar 

  6. Roland Becker and Rolf Rannacher. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4, 4:237–264, 1996.

    MathSciNet  MATH  Google Scholar 

  7. Roland Becker and Rolf Rannacher. An optimal control approach to a posteriori error estimation. In A. Iserles, editor, Acta  Numerica 2001, pages 1–102. Cambridge University Press, 2001.

    Google Scholar 

  8. Dietrich Braess. Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge University Press, Cambridge, 2007.

    MATH  Google Scholar 

  9. Heribert Blum and Franz-Theo Suttmeier. An adaptive finite element discretization for a simplified signorini problem. Calcolo, 37(2):65–77, 2000.

    MathSciNet  MATH  Google Scholar 

  10. Heribert Blum and Franz-Theo Suttmeier. Weighted error estimates for finite element solutions of variational inequalities. Computing, 65(2):119–134, 2000.

    MathSciNet  MATH  Google Scholar 

  11. Ivo Babuška and Theofanis Strouboulis. The Finite Element Method and its Reliability. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 2001.

    MATH  Google Scholar 

  12. Susanne C. Brenner and Larkin Ridgway Scott. The Mathematical Theory of Finite Element Methods. Springer Verlag, New York, 3. edition, 2008.

    MATH  Google Scholar 

  13. Roland Becker and Boris Vexler. A posteriori error estimation for finite element discretization of parameter identification problems. Numer. Math., 96:435–459, 2004.

    MathSciNet  MATH  Google Scholar 

  14. Roland Becker and Boris Vexler. Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations. J. Comp. Physics, 206(1):95–110, 2005.

    MathSciNet  MATH  Google Scholar 

  15. Olaf Benedix and Boris Vexler. A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl., 44(1):3–25, 2009.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Eduardo Casas and Joseph Frédéric Bonnans. Contrôle de systèmes elliptiques semilinéares comportant des contraintes sur l’état. In Nonlinear Partial Differential Equations and their Applications. Collège de France seminar, Vol. VIII (Paris, 1984–1985), volume 166 of Pitman Res. Notes Math. Ser., pages 69–86. Longman Sci. Tech., Harlow, 1988.

    Google Scholar 

  18. Eduardo Casas and Luis Alberto Fernández. Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim., 27:35–56, 1993.

    MathSciNet  MATH  Google Scholar 

  19. Graham F. Carey and J. Tinsley Oden. Finite Elements. Computational Aspects, volume 3. Prentice-Hall, 1984.

    Google Scholar 

  20. Carsten Carstensen and Rüdiger Verfürth. Edge residuals dominate a posteriori error estimates for low-order finite element methods. SIAM J. Numer. Anal., 36(5):1571–1587, 1999.

    MathSciNet  MATH  Google Scholar 

  21. Klaus Deckelnick, Andreas Günther, and Michael Hinze. Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math., 111:335–350, 2008.

    MathSciNet  MATH  Google Scholar 

  22. Kenneth Eriksson, Don Estep, Peter Hansbo, and Claes Johnson. Computational Differential Equations. Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

  23. The finite element toolkit Gascoigne. http://www.gascoigne.uni-hd.de.

  24. Andreas Günther and Michael Hinze. A posteriori error control of a state constrained elliptic control problem. J. Numer. Math., 16:307–322, 2008.

    MathSciNet  MATH  Google Scholar 

  25. Alexandra Gaevskaya, Roland H.W. Hoppe, Yuri Iliash, and Michael Kieweg. Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. In Control of Coupled Partial Differential Equations, International Series of Numerical Mathematics. Birkhäuser, 2007.

    MATH  Google Scholar 

  26. Andreas Günther and Anton Schiela. An interior point algorithm with inexact step computation in function space for state constrained optimal control. 35 pp. To appear in Numerische Mathematik, doi:10.1007/s00211-011-0381-4, 2011.

    Google Scholar 

  27. Andreas Günther andMoulay Hicham Tber. A goal-oriented adaptive moreauyosida algorithm for control- and state-constrained elliptic control problems. Preprint SPP1253-089, DFG Priority Program 1253, 2009.

    Google Scholar 

  28. Michael Hintermüller and Michael Hinze. Moreau-Yosida regularization in state constrained elliptic control problems: Error estimates and parameter adjustment. SIAM J. Numer. Anal., 47:1666–1683, 2008.

    MathSciNet  MATH  Google Scholar 

  29. Michael Hintermüller and Ronald H.W. Hoppe. Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim., 47(4):1721–1743, 2008.

    MathSciNet  MATH  Google Scholar 

  30. Michael Hintermüller, Ronald H.W. Hoppe, Yuri Iliash, and Michael 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.

    MathSciNet  MATH  Google Scholar 

  31. Ronald H.W. Hoppe, Yuri Iliash, Chakradhar Iyyunni, and Nasser H. Sweilam. A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Math., 14(1):57–82, 2006.

    MathSciNet  MATH  Google Scholar 

  32. Michael Hinze. A variational discretization concept in control constrained optimization: The linear-quadratic case. Comp. Optim. Appl., 30(1):45–61, 2005.

    MathSciNet  MATH  Google Scholar 

  33. Michael Hintermüller and Karl Kunisch. PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim., 20(3):1133–1156, 2009.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. Michael Hinze and Anton Schiela. Discretization of interior point methods for state constrained elliptic optimal control problems: Optimal error estimates and parameter adjustment. Comput. Optim. Appl., 48(3):581–600, 2011.

    MathSciNet  MATH  Google Scholar 

  36. Wenbin Liu. Adaptive multi-meshes in finite element approximation of optimal control. Contemporary Mathematics, (383):113–132, 2005.

    MathSciNet  MATH  Google Scholar 

  37. Ruo Li, Wenbin Liu, Heping Ma, and Tao Tang. Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim., 41(5):1321–1349, 2002.

    MathSciNet  MATH  Google Scholar 

  38. Freerk Auke Lootsma. Hessian matrices of penalty functions for solving constrained-optimization problems. Philips Res. Repts., 24:322–330, 1969.

    MathSciNet  MATH  Google Scholar 

  39. Wenbin Liu and Ningning Yan. A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math., 15(1):285–309, 2001.

    MathSciNet  MATH  Google Scholar 

  40. Dominik Meidner. Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems. PhD thesis, Mathematisch-Naturwissenschaftliche Gesamtfakultät, Universität Heidelberg, 2007.

    Google Scholar 

  41. Christian Meyer and Arnd Rösch. Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43(3):970–985, 2004.

    MathSciNet  MATH  Google Scholar 

  42. Walter Murray. Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions. J. Optim. Theory Appl., 7(3):189–196, 1971.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. RoDoBo: A C++ library for optimization with stationary and nonstationary PDEs. http://www.rodobo.uni-hd.de.

  45. Michael Schmich and Boris Vexler. Adaptivity with dynamic meshes for spacetime finite element discretizations of parabolic equations. SIAM J. Sci. Comput., 30(1):369–393, 2008.

    MATH  Google Scholar 

  46. Fredi Tröltzsch. Optimale Steuerung partieller Differentialgleichungen. Vieweg, 1. edition, 2005.

    Google Scholar 

  47. Rüdiger Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, New York-Stuttgart, 1996.

    MATH  Google Scholar 

  48. Boris Vexler. Adaptive Finite Element Methods for Parameter Identification Problems. PhD thesis, Mathematisch-Naturwissenschaftliche Gesamtfakultät, Universität Heidelberg, 2004.

    Google Scholar 

  49. VisuSimple: An interactive VTK-based visualization and graphics/mpeggeneration program. http://www.visusimple.uni-hd.de.

  50. Boris Vexler and Winnifried Wollner. Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim., 47(1):509–534, 2008.

    MathSciNet  MATH  Google Scholar 

  51. Winnifried 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(3):133–159, 2010.

    MathSciNet  MATH  Google Scholar 

  52. Winnifried Wollner. Adaptive Methods for PDE based Optimal Control with Pointwise Inequality Constraints. PhD thesis, Mathematisch-Naturwissenschaftliche Gesamtfakultät, Universität Heidelberg, 2010.

    Google Scholar 

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Author information

Authors and Affiliations

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

    Rolf Rannacher

  2. Lehrstuhl für Mathematische Optimierung Zentrum Mathematik, M1, Technische Universität München, Boltzmannstr. 3, D-85748, Garching bei München, Germany

    Boris Vexler

  3. Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

    Winnifried Wollner

Authors
  1. Rolf Rannacher
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  2. Boris Vexler
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  3. Winnifried Wollner
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Corresponding author

Correspondence to Rolf Rannacher .

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

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Rannacher, R., Vexler, B., Wollner, W. (2012). A Posteriori Error Estimation in PDE-constrained Optimization with Pointwise Inequality Constraints. 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_19

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