Advertisement

Computing and Visualization in Science

, Volume 13, Issue 5, pp 207–220 | Cite as

A practical framework for the construction of prolongation operators for multigrid based on canonical basis functions

  • Roman Wienands
  • Harald Köstler
Article
  • 105 Downloads

Abstract

We discuss a general framework for the construction of prolongation operators for multigrid methods. It turns out that classical black-box prolongation or prolongation operators based on smoothed aggregation can be classified as special cases. The approach is suitable both for geometric and for purely algebraic multigrid settings. It allows for a simple and efficient implementation and parallelization by introducing canonical basis functions. We show numerical results for several diffusion problems with strongly varying or jumping coefficients. As one possible application for our method we choose three-dimensional medical image segmentation. In addition to that a nonsymmetric convection-diffusion problem is presented.

Keywords

Multigrid Prolongation Matrix-dependent transfer operators Jumping coefficients Image segmentation Nonsymmetric convection-diffusion problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alcouffe R., Brandt A., Dendy J., Painter J.: The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comput. 2, 430–454 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Braess D.: Towards algebraic multigrid for elliptic prooblems of second order. Computing 55, 379–393 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brandt A.: General highly accurate algebraic coarsening. Electronic Trans. Numer. Anal. 10, 1–20 (2000)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Brandt A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Brandt, A.: 1984 multigrid guide with applications to fluid dynamics. Monograph, GMD-Studie 85, GMD-FIT, Postfach 1240, D-5205, St. Augustin 1, West Germany (1985)Google Scholar
  6. 6.
    Brandt A., Yavneh I.: Accelerated multigrid convergence and high-reynolds recirculating flows. SIAM J. Sci. Comput. 14(3), 607–626 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brezina M., Cleary A., Falgout R., Henson V., Jones J., Manteuffel T., McCormick S., Ruge J.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22(5), 1570–1592 (2001)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Brezina M., Falgout R., MacLachlan S., Manteuffel T., McCormick S., Ruge J.: Adaptive smoothed aggregation (alphaSA). SIAM J. Sci. Comput. 25(6), 1896–1920 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Briggs W. L., Henson V. E., McCormick S. F.: A Multigrid Tutorial. 2nd edn. SIAM, Philadelphia (2000)zbMATHGoogle Scholar
  10. 10.
    de Zeeuw P.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33, 1–27 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dendy J. E. Jr.: Black box multigrid. J. Comput. Phys. 48, 366–386 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dendy J. E. Jr.: Black box multigrid for nonsymmetric problems. Appl. Math. Comput. 13, 261–284 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dendy J. E. Jr.: Two multigrid methods for three-dimensional equations with highly discontinuous coefficients. SIAM J. Sci. Stat. Comput. 8, 673–685 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Falgout R.D.: Introduction to algebraic multigrid. Comput. Sci. Eng. 8(6), 24–33 (2006)CrossRefGoogle Scholar
  15. 15.
    Grady L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
  16. 16.
    Grady, L., Tasdizen, T., Whitaker, R.: A geometric multigrid approach to solving the 2D inhomogeneous Laplace equation with internal Dirichlet boundary conditions. In: IEEE International Conference on Image Processing (ICIP), vol. 2, 2005Google Scholar
  17. 17.
    Hackbusch W.: Multi-Grid Methods Applications. Springer, Berlin (1985)zbMATHGoogle Scholar
  18. 18.
    Hackbusch, W., Trottenberg, U. (eds): Multigrid Methods. Springer, Berlin (1982)zbMATHGoogle Scholar
  19. 19.
    Han J., Bennewitz C., Köstler H., Hornegger J., Kuwert T.: Computer-aided validation of hybrid SPECT/CT scanners. Comput. Med. Imaging Graph. 32(5), 388–395 (2008)CrossRefGoogle Scholar
  20. 20.
    Kimmel R., Yavneh I.: An algebraic multigrid approach for image analysis. SIAM J. Sci. Comput. 24(4), 1218–1231 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Köstler, H.: A Multigrid Framework for Variational Approaches in Medical Image Processing and Computer Vision. Verlag Dr. Hut, München (2008)Google Scholar
  22. 22.
    Ruge J. W., Stüben K.: Algebraic multigrid (AMG). In: McCormick, S.F. (eds) Multigrid Methods, Frontiers in Applied Mathematics, pp. 73–130. SIAM, Philadelphia (1987)Google Scholar
  23. 23.
    Stüben, K.: Algebraic multigrid (AMG): An introduction with applications. GMD-Forschungszentrum Informationstechnik, appeared as an appendix in [24] (1999)Google Scholar
  24. 24.
    Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, London and San Diego (2001)zbMATHGoogle Scholar
  25. 25.
    Vaněk P., Brezina M., Mandel J.: Convergence of algebraic multigrid based on smoothed aggregation. Numerische Mathematik 88(3), 559–579 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Vaněk P., Mandel J., Brezina M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56(3), 179–196 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Vassilevski P.: Element-free AMGe: General algorithms for computing interpolation weights in AMG. SIAM J. Sci. Comput. 23(2), 629–650 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wagner C.: On the algebraic construction of multilevel transfer operators. Computing 65(1), 73–95 (2000)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Wesseling P.: An Introduction to Multigrid Methods, Ser. Pure and Applied Mathematics. Wiley, Chichester (1992)Google Scholar
  30. 30.
    Wienands R., Yavneh I.: Collocation coarse approximation (CCA) in multigrid. SIAM J. Sci. Comput. 31, 3643–3660 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Yavneh I.: Coarse-grid correction for nonelliptic and singular perturbation problems. SIAM J. Sci. Comput. 19, 1682 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Yavneh I., Venner C., Brandt A.: Fast multigrid solution of the advection problem with closed characteristics. SIAM J. Sci. Comput. 19, 111 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Yavneh I.: Why multigrid methods are so efficient. Comput. Sci. Eng. 8(6), 12–22 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of CologneCologneGermany
  2. 2.Lehrstuhl Informatik 10 (Systemsimulation)Universität Erlangen-NurembergErlangenGermany

Personalised recommendations