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Two-Sided Guaranteed Estimates of the Cost Functional for Optimal Control Problems with Elliptic State Equations

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Optimization with PDE Constraints

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 101))

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Abstract

In the paper, we discuss error estimation methods for optimal control problems with distributed control functions entering the right-hand side of the corresponding elliptic state equations. Our analysis is based on a posteriori error estimates of the functional type, which were derived in the last decade for many boundary value problems. They provide guaranteed two-sided bounds of approximation errors for any conforming approximation. If they are applied to approximate solutions of state equations, then we obtain new variational formulations of optimal control problems and guaranteed bounds of the cost functional. Moreover, for problems with linear state equations this procedure leads to guaranteed and computable error estimates for the state and control functions.

Mathematics Subject Classification (2010). Primary 65K15; Secondary 49M99, 65K15.

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References

  1. V. Arnautu, P. Neittaanmäki, Optimal Control from Theory to Computer Programs (Kluwer, Dordrecht, 2003)

    Book  MATH  Google Scholar 

  2. R. Becker, R. Rannacher, An optimal control approach to error estimation and mesh adaptation, in Acta Numerica 2000, ed. by A. Iserles (Cambridge University Press, Cambridge, 2001), pp. 1–102.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  MathSciNet  MATH  Google Scholar 

  5. 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, International series of numerical mathematics, vol. 155 (Birkhäuser, Basel, 2007), pp. 47–68

    Google Scholar 

  6. A. Gaevskaya, W.H. Hoppe, S. Repin, A posteriori error estimation for elliptic optimal control problems with distributed control. J. Math. Sci. 144, 1–14 (2007)

    Article  MathSciNet  Google Scholar 

  7. M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Q. Appl. Math. LXI, 131–161 (2003)

    Google Scholar 

  8. R.H.W. Hoppe, Y. Iliash, C. 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  MathSciNet  MATH  Google Scholar 

  9. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, Berlin/Heidelberg/New York, 1971)

    Book  MATH  Google Scholar 

  10. W.B. Liu, P. Neittaanmäki, D. Tiba, Existence for shape optimization problems in arbitrary dimension. SIAM J. Control Optim. 41, 1440–1454 (2002)

    Article  MathSciNet  Google Scholar 

  11. O. Mali, P. Neittaanäki, S. Repin, Accuracy Verification Methods. Theory and Algorithms (Springer, Berlin/Heidelberg/New York, 2013)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. S.G. Mikhlin, Variational Methods in Mathematical Physics (Pergamon Press, Oxford, 1964)

    MATH  Google Scholar 

  14. P. Neittaanmäki, S. Repin, Reliable Methods for Computer Simulation, Error Control and A Posteriori Estimates (Elsevier, New York, 2004)

    MATH  Google Scholar 

  15. P. Neittaanmäki, D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications (Marcel Dekker, New York, 1994)

    MATH  Google Scholar 

  16. P. Neittaanmäki, D. Tiba, Fixed domain approaches in shape optimization problems. Inverse Probl. 28 (2012). doi:10.1088/0266-5611/28/9/093001

    Google Scholar 

  17. P. Neittaanmäki, J. Sprekls, D. Tiba, Optimization of Elliptic Systems. Theory and Applications (Springer, Berlin/Heidelberg/New York, 2006)

    Google Scholar 

  18. P. Neittaanmäki, A. Pennanen, D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions. Inverse Probl. 25, 055003 (2009)

    Article  Google Scholar 

  19. L.E. Payne, H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauchn, Semin, V.A. Steklov Mathematical Institute in St.-Petersburg (POMI) 243, 201–214 (1997)

    Google Scholar 

  21. S. Repin, A posteriori error estimates for variational problems with uniformly convex functionals. Math. Comput. 69, 481–500 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations. Proc. St. Petersburg Math. Society IX, 143–171 (2001) [Translation in Am. Math. Soc. Transl. Ser. 2 209. American Mathematical Society, Providence, 2003]

    Google Scholar 

  23. S. Repin, A Posteriori Estimates for Partial Differential Equations (De Gruyter, Berlin, 2008)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department Mathematical Information Technology, University of Jyväskylä, 35, (Agora), FI-40014, Jyväskylä, Finland

    Pekka Neittaanmäki & Sergey Repin

Authors
  1. Pekka Neittaanmäki
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  2. Sergey Repin
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Correspondence to Sergey Repin .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Ronald Hoppe

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Neittaanmäki, P., Repin, S. (2014). Two-Sided Guaranteed Estimates of the Cost Functional for Optimal Control Problems with Elliptic State Equations. In: Hoppe, R. (eds) Optimization with PDE Constraints. Lecture Notes in Computational Science and Engineering, vol 101. Springer, Cham. https://doi.org/10.1007/978-3-319-08025-3_10

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