, Volume 84, Issue 1–2, pp 27–48 | Cite as

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

  • O. Lass
  • M. Vallejos
  • A. BorziEmail author
  • C. C. Douglas


The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.


Elliptic optimal control problem Finite elements Multigrid method 

Mathematics Subject Classification (2000)

49K20 65N12 65N30 65N55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borzì A, Schulz V (2009) Multigrid methods for pde optimization. SIAM Rev (to appear)Google Scholar
  2. 2.
    Borzì A (2003) Multigrid methods for parabolic distributed optimal control problems. J Comput Appl Math 157: 365–382zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borzì A (2005) On the convergence of the MG/OPT method. PAMM 5: 735–736CrossRefGoogle Scholar
  4. 4.
    Borzì A (2007) High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems. J Comput Appl Math 200: 67–85zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Borzì A, Kunisch K (2005) A multigrid scheme for elliptic constrained optimal control problems. Comput Optim Appl 31: 309–333zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borzì A, Kunisch K, Kwak DY (2002) Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J Control Optim 41: 1477–1497CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bramble JH, Pasciak JE, Xu J (1991) The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math Comp 56: 1–34zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bramble JH, Kwak DY, Pasciak JE (1994) Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J Numer Anal 31: 1746–1763zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bramble JH (1993) Multigrid methods. Pitman research notes in mathematics series. EssexGoogle Scholar
  10. 10.
    Brandt A (1977) Multi-level adaptive solutions to boundary-value problems. Math Comp 31: 333–390zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brenner SC, Scott LR (2008) The mathematical theory of finite element methods. Springer, New YorkzbMATHGoogle Scholar
  12. 12.
    Douglas CC (1995) Madpack: a family of abstract multigrid or multilevel solvers. Comput Appl Math 14: 3–20zbMATHGoogle Scholar
  13. 13.
    Douglas CC (1984) Multi-grid algorithms with applications to elliptic boundary-value problems. SIAM J Numer Anal 21: 236–254zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dreyer T, Maar B, Schulz V (2000) Multigrid optimization in applications. SQP-based direct discretization methods for practical optimal control problems. J Comput Appl Math 120: 67–84zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hackbusch W (1980) Fast solution of elliptic control problems. J Optim Theory Appl 31: 565–581zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hackbusch W (1985) Multigrid methods and applications. Springer, BerlinGoogle Scholar
  17. 17.
    Jung M, Langer U (2008) Methode der finiten elemente für ingenieure. Teubner, StuttgartGoogle Scholar
  18. 18.
    Lewis RM, Nash S (2005) Model problems for the multigrid optimization of systems governed by differential equations. SIAM J Sci Comput 26: 1811–1837zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Nash S (2000) A multigrid approach to discretized optimization problems. Optim Methods Softw 14: 99–116zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New YorkzbMATHCrossRefGoogle Scholar
  21. 21.
    Oh S, Bouman C, Webb KJ (2006) Multigrid tomographic inversion with variable resolution data and image spaces. IEEE Trans Image Process 15: 2805–2819CrossRefGoogle Scholar
  22. 22.
    Oh S, Milstein A, Bouman C, Webb KJ (2001) Multigrid algorithms for optimization and inverse problems. IEEE Trans Image Process 10: 909–922CrossRefGoogle Scholar
  23. 23.
    Trottenberg U, Oosterlee CW, Schüller A (2001) Multigrid. Academic Press, San DiegozbMATHGoogle Scholar
  24. 24.
    Vallejos M, Borzì A (2008) Multigrid optimization methods for linear and bilinear elliptic optimal control problems. J Comput 82: 31–52zbMATHCrossRefGoogle Scholar
  25. 25.
    Ye JC, Bouman C, Webb KJ, Millane R (2001) Nonlinear multigrid algorithms for bayesian optical diffusion tomography. IEEE Trans Image Process 10: 909–922zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • O. Lass
    • 1
  • M. Vallejos
    • 1
    • 2
  • A. Borzi
    • 1
    • 3
    Email author
  • C. C. Douglas
    • 4
  1. 1.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-UniversitätGrazAustria
  2. 2.Institute of Mathematics, College of ScienceUniversity of the PhilippinesDilimanPhilippines
  3. 3.Dipartimento e Facoltà di IngegneriaUniversità degli Studi del SannioBeneventoItaly
  4. 4.Mathematics DepartmentUniversity of WyomingLaramieUSA

Personalised recommendations