Computing and Visualization in Science

, Volume 7, Issue 3–4, pp 183–188 | Cite as

An efficient algebraic multigrid method for solving optimality systems

Regular article

Abstract

An algebraic multigrid method (AMG) for solving convection-diffusion optimality systems is presented. Results of numerical experiments demonstrate robustness of the AMG scheme with respect to changes of the weight of the cost of the control and show that the computational performance of the proposed AMG scheme is comparable to that of AMG applied to single scalar equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arian, E., Ta’asan, S.: Smoothers for optimization problems. In Seventh Copper Mountain Conference on Multigrid Methods, Vol. CP3339, NASA Conference Publication, NASA, Duane Melson, N., Manteuffel, T.A., McCormick, S.F., Douglas, C.C. (eds.), Hampton, VA, 1995, pp. 15–30Google Scholar
  2. 2.
    Bertsekas, D.P.: Nonlinear Programming. Belmont: Athena Scientific 1995Google Scholar
  3. 3.
    Borzì, A., Borzì, G.: An algebraic multigrid method for a class of elliptic differential systems. SIAM J. Sci. Comp. 25(1), 302–323 (2003)CrossRefGoogle Scholar
  4. 4.
    Borzì, A., Kunisch, K.: The numerical solution of the steady state solid fuel ignition model and its optimal control. SIAM J. Sci. Comp. 22(1), 263–284 (2000)CrossRefGoogle Scholar
  5. 5.
    Borzì, A., Kunisch, K., Kwak, D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41(5), 1477–1497 (2003)CrossRefGoogle Scholar
  6. 6.
    Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing 55, 379–393 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Mathematics of Computation 56, 1–34 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Mathematics of Computation 57, 23–45 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bramble, J.H., Kwak, D.Y., Pasciak, J.E.: Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal. 31, 1746–1763 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brandt, A.: Algebraic multigrid theory: The symmetric case. Proc. Int. Multigrid Conf., Copper Mountain, Colorado, 1983; Appl. Math. Comp. 19, 23–56 (1986)Google Scholar
  11. 11.
    Brandt, A.: General highly algebraic coarsening. ETNA 10, 1–20 (2000)Google Scholar
  12. 12.
    Brezina, M., Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22(5), 1570–1592 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dreyer, Th., Maar, B., Schulz, V.: Multigrid optimization in applications. J. Comput. Appl. Math. 120, 67–84 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hackbusch, W.: Fast solution of elliptic control problems. Journal of Optimization Theory and Application 31, 565–581 (1980)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. New York: Springer-Verlag 1994Google Scholar
  16. 16.
    Hoffmann, K.H., Hoppe, R., Schulz, V. (eds.): Fast Solution of Discretized Optimization Problems. International Series on Numerical Mathematics, Vol. 138, Birkhäuser 2001Google Scholar
  17. 17.
    Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer 1971Google Scholar
  18. 18.
    Mandel, J.: Local Approximation Estimators for Algebraic Multigrid. ETNA 15, 56–65 (2003)MathSciNetGoogle Scholar
  19. 19.
    Muszynski, P., Rüde, U., Zenger, Chr.: Application of algebraic multigrid (AMG) to constrained quadratic optimization. technical report TUM-I8801, Technische Universität München, 1988Google Scholar
  20. 20.
    Ruge, J.W., Stüben, K.: Algebraic Multigrid (AMG). In: McCormick, S. (ed.), Multigrid Methods, Frontiers in Applied Mathematics, Vol. 5, Philadelphia: SIAM 1986Google Scholar
  21. 21.
    Saad, Y.: SPARSKIT: A basic tool kit for sparse matrix computations. Rep. No. 90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, CA, 1990Google Scholar
  22. 22.
    Schulz, V., Wittum, G.: Multigrid optimization methods for stationary parameter identification problems in groundwater flow. In: Hackbusch, W., Wittum, G. (eds.): Multigrid Methods V, Lecture Notes in Computational Science and Engineering 3, pp. 276–288, Springer 1998Google Scholar
  23. 23.
    Stüben, K.: Algebraic Multigrid (AMG): An Introduction with Applications. GMD Report 53, March 1999Google Scholar
  24. 24.
    Vanek, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numerische Mathematik 88(3), 559–579 (2001)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria
  2. 2.Department of MathematicsUniversity of MessinaMessinaItaly

Personalised recommendations