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Journal of Optimization Theory and Applications

, Volume 119, Issue 3, pp 499–533 | Cite as

Augmented Lagrangian Active Set Methods for Obstacle Problems

  • T. Kärkkäinen
  • K. Kunisch
  • P. Tarvainen
Article

Abstract

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.

Obstacle problems active set strategies augmented Lagrangian methods multigrid and multilevel methods 

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References

  1. 1.
    Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer Verlag, New York, NY, 1984.Google Scholar
  2. 2.
    Glowinski, R., Lions, J. L., and Tremolieres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.Google Scholar
  3. 3.
    Hackbusch, W., and Mittelmann, H., On Multigrid Methods for Variational Inequalities, Numerische Mathematik, Vol. 42, pp. 65-76, 1983.Google Scholar
  4. 4.
    Hoffmann, K. H., and Zou, J., Parallel Algorithms of Schwarz Variant for Variational Inequalities, Numerical Functional Analysis and Optimization, Vol. 13, pp. 449-462, 1992.Google Scholar
  5. 5.
    Kornhuber, R., Monotone Multigrid Methods for Elliptic Variational Inequalities, I, Numerische Mathematik, Vol. 69, pp. 167-184, 1994.Google Scholar
  6. 6.
    Lions, P. L., On the Schwarz Alternating Method, I, 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations, Edited by R. Glowinski, G. Golub, G. Meurant, and J. Periaux, SIAM, Philadelphia, Pennsylvania, pp. 2-42, 1988.Google Scholar
  7. 7.
    SchÖberl, J., Solving the Signorini Problem on the Basis of Domain Decomposition Techniques, Computing, Vol. 60, pp. 323-344, 1998.Google Scholar
  8. 8.
    Tarvainen, P., Two-Level Schwarz Method for Unilateral Variational Inequalities, IMA Journal of Numerical Analysis, Vol. 19, pp. 193-212, 1999.Google Scholar
  9. 9.
    Zeng, J., and Zhou, S., On Monotone and Geometric Convergence of Schwarz Methods for Two-Sided Obstacle Problems, SIAM Journal on Numerical Analysis, Vol. 35, pp. 600-616, 1998.Google Scholar
  10. 10.
    Tarvainen, P., Numerical Algorithms Based on Characteristic Domain Decomposition for Obstacle Problems, Communications in Numerical Methods in Engineering, Vol. 13, pp. 793-801, 1997.Google Scholar
  11. 11.
    Hoppe, R., Multigrid Algorithms for Variational Inequalities, SIAM Journal on Numerical Analysis, Vol. 24, pp. 1046-1065, 1987.Google Scholar
  12. 12.
    Hoppe, R., Two-Sided Approximations for Unilateral Variational Inequalities by Multigrid Methods, Optimization, Vol. 18, pp. 867-881, 1987.Google Scholar
  13. 13.
    Kirsten, H., and Tichatschke, R., On a Method of Feasible Directions for Solving Variational Inequalities, Optimization, Vol. 16, pp. 535-546, 1985.Google Scholar
  14. 14.
    Lions, P. L., and Mercier, B., Approximation Numérique des É quations de Hamilton-Jacobi-Bellman, RAIRO Analyse Numérique, Vol. 14, pp. 375-387, 1980.Google Scholar
  15. 15.
    Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.Google Scholar
  16. 16.
    Ito, K., and Kunisch, K., Augmented Lagrangian Methods for Nonsmooth, Convex Optimization in Hilbert Spaces, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 41, pp. 591-616, 2000.Google Scholar
  17. 17.
    Hoppe, R., and Kornhuber, R., Adaptive Multilevel Methods for Obstacle Problems, SIAM Journal on Numerical Analysis, Vol. 31, pp. 301-323, 1994.Google Scholar
  18. 18.
    KÄrkkÄinen, T., and Toivanen, J., Multigrid Methods with Extended Subspaces for Reduced Systems, Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applications, Edited by P. NeittaanmÄki, T. Tiihonen, and P. Tarvainen, World Scientific, Singapore, pp. 564-570, 2000.Google Scholar
  19. 19.
    KÄrkkÄinen, T., and Toivanen, J., Building Blocks for Odd-Even Multigrid with Applications to Reduced Systems, Journal of Computational and Applied Mathematics, Vol. 131, pp. 15-33, 2001.Google Scholar
  20. 20.
    KÄrkkÄinen, T., Kunisch, K., and Tarvainen, P., Primal-Dual Active Set Methods for Obstacle Problems, Report B2, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2000.Google Scholar
  21. 21.
    Varga, R., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.Google Scholar
  22. 22.
    Windisch, G., M-Matrices in Numerical Analysis, Teubner, Leipzig, Germany, 1989.Google Scholar
  23. 23.
    Ito, K., and Kunisch, K., An Augmented Lagrangian Technique for Variational Inequalities, Applied Mathematics and Optimization, Vol. 21, pp. 223-241, 1990.Google Scholar
  24. 24.
    Kornhuber, R., and Yserentant, H., Multilevel Methods for Elliptic Problems on Domains Not Resolved by the Coarse Grid, Domain Decomposition Methods in Scientific and Engineering Computing, Edited by D. E. Keyes, and J. Xu, Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, Vol 180, pp. 49-60, 1993.Google Scholar
  25. 25.
    Hackbusch, W., Multigrid Methods and Applications, Springer Series in Computational Mathematics, Springer Verlag, Berlin, Germany, 1985.Google Scholar
  26. 26.
    Dostal, Z., Box-Constrained Quadratic Programming with Proportioning and Projections, SIAM Journal on Optimization, Vol. 7, pp. 871-887, 1997.Google Scholar
  27. 27.
    Rodrigues, J. F., Obstacle Problems in Mathematical Physics, North-Holland, New York, NY, 1987.Google Scholar
  28. 28.
    Bramble, J., Pasciak, J., and Xu, J., Parallel Multilevel Preconditioners, Mathematics of Computation, Vol. 55, pp. 1-22, 1990.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • T. Kärkkäinen
    • 1
  • K. Kunisch
    • 2
  • P. Tarvainen
    • 3
  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.Institute for MathematicsUniversity of GrazGrazAustria
  3. 3.JyväskyläFinland

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