Abstract
We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.









Similar content being viewed by others
References
Ainsworth, M., Oden, J.: A Posteriori error estimation in finite element analysis. Pure and applied mathematics: a Wiley series of texts, monographs and tracts. Wiley (2000)
Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978)
Babuška, I., Strouboulis, T., Whiteman, J.: Finite Elements: An Introduction to the Method and Error Estimation. Oxford University Press, Oxford (2011)
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Partial Differential Equations. No. 1 in Lectures in Mathematics. ETH Zürich. Birkhäuser (2003)
Bartels, S., Carstensen, C.: Each averaging technique yields reliable a posteriori control in FEM on unstructured grids. Part I: Low-order conforming, non-conforming and mixed FEM. Math. Comp. 71(239), 945–969 (2002)
Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39(1), 113–132 (2000)
Benedix, O., Vexler, B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Computat. Optim. Appl. 44(1), 3–25 (2009)
Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)
Bergounioux, M., Haddou, M., Hintermüller, M., Kunisch, K.: A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11(2), 495–521 (2000)
Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999)
Bonito, A., Nochetto, R.: Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal. 48(2), 734–771 (2010)
Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)
Casas, E., Mateos, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 3(20), 803–822 (2014)
Cherednichenko, S., Krumbiegel, K., Rösch, A.: Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems 24(6), (2008)
Ciarlet, P.: The finite element method for elliptic problems. SIAM classics in applied mathematics, Philadelphia (2002)
Gaspoz, F., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009)
Günther, A., Hinze, M.: A posteriori error control of a state-constrained elliptic control problem. J. Numer. Math. 16, 307–322 (2008)
Hintermüller, M., Hinze, M.: Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47(3), 1666–1683 (2009)
Hintermüller, M., Hoppe, R.: Goal-oriented adaptivity in control-constrained optimal control of partial differential equations. SIAM J. Control Optim. 47(4), 1721–1743 (2008)
Hintermüller, M., Hoppe, R.: Goal-oriented mesh adaptivity for mixed control-state constrained elliptic optimal control problems. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Périaux, J., Pironneau O. (eds.) Applied and Numerical Partial Differential Equations, Comput. Methods Appl. Sci., vol. 15, pp. 97–111. Springer, Berlin (2010)
Hintermüller, M., Hoppe, R., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic optimal control problems with control constraints. ESAIM Control Optim. Calc. Var. 14(3), 865–888 (2008)
Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006)
Hintermüller, M., Schiela, A., Wollner, W.: The length of the primal-dual path in Moreau-Yosida-based path-following methods for state constrained optimal control. SIAM J. Optim 24(1), 108–126 (2014)
Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005)
Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic control problems. Comput. Optim. Appl. 46(3), 487–510 (2010)
Hinze, M., Schiela, A.: 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)
Hoppe, R., Kieweg, M.: Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput. Optim. Appl. 46(3), 511–533 (2010)
Kohls, K., Rösch, A., Siebert, K.: A posteriori error analysis of optimal control problems with control constraints. SIAM J. Cont. Optim. 52(3), 1832–1861 (2014)
Kossacký, I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55(3), 275–288 (1994)
Krumbiegel, K., Rösch, A.: On the regularization error of state constrained Neumann control problems. Control Cybernet. 37(2), 369–392 (2008)
Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43(2), 213–233 (2012)
Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13(2), 321–334 (2002)
Kurcyusz, S., Zowe, J.: Regularity and stability for the mathematical programming problem in Banach spaces. Appli. Math. Optim. 5(1), 49–62 (1979)
Li, R., Liu, W., Yan, N.: A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 33(2), 155–182 (2007)
Liu, W., Yan, N.: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math 15(1–4), 285–309 (2001)
Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybernet. 37(1), 51–85 (2008)
Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22(6), 871–899 (2007)
Meyer, C., Rademacher, A., Wollner, W.: Adaptive optimal control of the obstacle problem. SIAM J. Sci. Comp. 37(2), A918–A945 (2015)
Morin, P., Siebert, K.G., Veeser, A.: A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18(5), 707–737 (2008)
Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)
Rösch, A., Wachsmuth, D.: A posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120(4), 733–762 (2012)
Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim 20(2), 1002–1031 (2009)
Schiela, A., Wollner, W.: Barrier methods for optimal control problems with convex nonlinear gradient state constraints. SIAM J. Optim 21(1), 269–286 (2011)
Schmidt, A., Siebert, K.: Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA, vol. 42. Springer (2005)
Siebert, K.: A convergence proof for adaptive finite elements without lower bounds. IMA J. Numer. Anal. 31(3), 947–970 (2011)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second order à coefficients discontinus. Ann. Inst. Fourier 15(1), 189–257 (1965)
Steinig, S.: Adaptive finite elements for state-constrained optimal control problems - convergence analysis and a posteriori error estimation. Ph.D. thesis, Universität Stuttgart (2014)
Tröltzsch, F.: Optimal control of partial differential equations: theory, methods, and applications. Grad. Stud. Math . American Mathematical Society (2010)
Veeser, A., Verfürth, R.: Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal. 47(3), 2387–2405 (2009)
Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advanced Numerical Mathematics. Wiley, Chichester (1996)
Vexler, B., Wollner, W.: Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Contr. Opt. 47(1), 509–534 (2008)
Wollner, W.: 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(1), 133–159 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rösch, A., Siebert, K.G. & Steinig, S. Reliable a posteriori error estimation for state-constrained optimal control. Comput Optim Appl 68, 121–162 (2017). https://doi.org/10.1007/s10589-017-9908-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-017-9908-7