Skip to main content
SpringerLink
Log in

Approximation of sparse controls in semilinear equations by piecewise linear functions

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Semilinear elliptic optimal control problems involving the \(L^1\) norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comp. Optim. Appl. 23(2), 201–229 (2002)

    Article  MATH  Google Scholar 

  2. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994)

    MATH  Google Scholar 

  3. Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN Math. Model. Numer. Anal. 33(6), 1187–1202 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casas, E.: Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math. 26, 137–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E., Herzog, R., Wachsmuth, G.: Optimality conditions and error analysis of semilinear elliptic control problems with \(L^1\) cost functional. SIAM J. Optim. (in press)

  6. Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Casas, E., Mateos, M.: Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl. 39, 265–295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, E., Raymond, J.-P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45(5), 1586–1611 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, E., Tröltzsch, F.: Error estimates for linear-quadratic elliptic control problems. In: Barbu, V. et al. (eds.) Proceedings of the IFIP TC7/WG 7.2 International Working Conference on Analysis and Optimization of Differential Systems, September 10–14, 2002, Constanta, Romania. Analysis and Optimization of Differential Systems, pp. 89–100. Kluwer Academic Press, Boston (2003)

  10. CGAL: Computational Geometry Algorithms Library. http://www.cgal.org

  11. Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: Control Optim. Calc. Var. 17(1), 243–266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. de los Reyes, J.C., Meyer, C., Vexler, B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37(2), 251–284 (2008)

    MathSciNet  MATH  Google Scholar 

  13. FEniCS. FEniCS project. http://www.fenicsproject.org/ (2007)

  14. Herzog, R., Stadler, G., Wachsmuth, G.: Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50(2), 943–963 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Logg, A.: Automating the finite element method. Arch. Comput. Methods Eng. 14(2), 93–138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Meyer, C., Rösch, A.: \(L^\infty \)-estimates for approximated optimal control problems. SIAM J. Control Optim. 44(5), 1636–1649 (2005)

    Article  MathSciNet  Google Scholar 

  17. Raviart, P.A., Thomas, J.M.: Introduction à L’analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983)

    MATH  Google Scholar 

  18. Rösch, A.: Error estimates for linear-quadratic optimal control problems with control constraints. Optim. Methods Softw. 21(1), 121–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schatz, A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: part I. Global Estim. Math. Comp. 67(223), 877–899 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Stadler, G.: Elliptic optimal control problems with \({L}^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44(2), 159–181 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Vossen, G., Maurer, H.: On \(L^1\)-minimization in optimal control and applications to robotics. Optim. Control Appl. Methods 27(6), 301–321 (2006)

    Article  MathSciNet  Google Scholar 

  22. Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var. 17(3), 858–886 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, Av. Los Castros s/n, 39005,  Santander, Spain

    Eduardo Casas

  2. Faculty of Mathematics, Chemnitz University of Technology, Reichenhainer Straße 41, 09126,  Chemnitz, Germany

    Roland Herzog & Gerd Wachsmuth

Authors
  1. Eduardo Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roland Herzog
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gerd Wachsmuth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Herzog.

Additional information

E. Casas was partially supported by the Spanish Ministerio de Economía y Competitividad under the project MTM2011-22711.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Casas, E., Herzog, R. & Wachsmuth, G. Approximation of sparse controls in semilinear equations by piecewise linear functions. Numer. Math. 122, 645–669 (2012). https://doi.org/10.1007/s00211-012-0475-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-012-0475-7

Mathematics Subject Classification

Search

Navigation