Journal of Statistical Physics

, Volume 74, Issue 1–2, pp 313–348 | Cite as

Optimal multigrid algorithms for calculating thermodynamic limits

  • A. Brandt
  • M. Galun
  • D. Ron


Beyond eliminating the critical slowing down, multigrid algorithms can also eliminate the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced in coarse grids during the multigrid cycle. Thermodynamic limits can be calculated to accuracy ɛ in justO-2) computer operations. Examples described in detail and with results of numerical tests are the calculation of the susceptibility, the σ-susceptibility, and the average energy in Gaussian models, and also the determination of the susceptibility and the critical temperature in a two-dimensional Ising spin model. Extension to more advanced models is outlined.

Key Words

Multigrid Gaussian model Ising spin model XY model Monte Carlo thermodynamic limit coarsening by approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. E. Alcouffe, A. Brandt, J. E. Dendy, Jr., and J. W. Painter, The multi-grid methods for the diffusion equation with strongly discontinuous coefficients.SIAM J. Sci. Stat. Comp. 2:430–454 (1981).Google Scholar
  2. 2.
    A. Brandt, Multigrid Techniques: 1984 Guide, with Applications to Fluid Dynamics [available as GMD Studien Nr. 85, GMD-AIW, Postfach 1240, D-5205, St. Augustin 1, Germany].Google Scholar
  3. 3.
    A. Brandt, Multilevel computations: Reviews and recent developments, inPreliminary Proceedings 3rd Copper Mountain Conference on Multigrid Methods (April 1987); see also inMultigrid Methods: Theory Applications and Super-computing, S. F. McCormick, ed. (Marcel Dekker, New York, 1988), pp. 35–62.Google Scholar
  4. 4.
    A. Brandt, The Weizmann Institute research in multilevel computation: 1988 report, inProceedings 4th Copper Mountain Conference on Multigrid Methods, J. Mandelet al., eds. (SIAM, 1989), pp. 13–53.Google Scholar
  5. 5.
    A. Brandt, Multigrid methods in lattice field computations,Nucl. Phys. B. (Proc. Suppl.) 26:137–180 (1992).Google Scholar
  6. 6.
    A. Brandt, D. Ron, and D. J. Amit, Multi-level approaches to discrete-state and stochastic problems, inMultigrid Methods, W. Hackbusch and U. Trottenberg, eds. (Springer-Verlag, Berlin, 1986), pp. 66–99.Google Scholar
  7. 7.
    M. Galun, Optimal multigrid algorithms for model problems in statistical mechanics, M.Sc. Thesis, Weizmann Institute of Science (1992).Google Scholar
  8. 8.
    J. Goodman and A. D. Sokal, Multigrid Monte Carlo methods for lattice field theories,Phys. Rev. Lett. 56:1015–1018 (1986).Google Scholar
  9. 9.
    S. Gottlieb, W. Liu, D. Toussaint, and R. L. Sugar, Testing an exact algorithm for simulation of fermionic QCD,Phys. Rev. D 35:2611 (1987).Google Scholar
  10. 10.
    D. Kandel and E. Domany, General cluster Monte Carlo dynamics,Phys. Rev. B. 43:8539 (1991).Google Scholar
  11. 11.
    D. Kandel, E. Domany, and A. Brandt, Simulations without critical slowing down—Ising and 3-state Potts models.Phys. Rev. B 40:330 (1989).Google Scholar
  12. 12.
    D. Kandel, E. Domany, D. Ron, A. Brandt, and E. Loh, Jr., Simulations without critical slowing down,Phys. Rev. Lett. 60:1591 (1988).Google Scholar
  13. 13.
    X.-J. Li and A. D. Sokal, Rigorous lower bound on the dynamic critical exponent of some multilevel Swendsen-Wang algorithms,Phys. Rev. Lett. 67:1482 (1991).Google Scholar
  14. 14.
    G. Mack and A. Pordt, Convergent perturbation expansions for Euclidean quantum field theory,Commun. Math. Phys. 97:267 (1985); G. Mack, inNonperturbative Quantum Field Theory, G. t'Hooftet al.,eds. (Plenum Press, New York, 1988), p. 309.Google Scholar
  15. 15.
    D. Ron, Development of fast numerical solvers for problems in optimization and statistical mechanics, Ph.D. Thesis, Weizmann Institute of Science (1989).Google Scholar
  16. 16.
    S. Shmulyian, Multilevel Monte Carlo algorithms for spin models, M.Sc. Thesis, Weizmann Institute of Science (1993).Google Scholar
  17. 17.
    A. D. Sokal, How to beat critical slowing-down: 1990 update,Nucl. Phys. B (Proc. Suppl.) 20:55–67 (1991).Google Scholar
  18. 18.
    R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations,Phys. Rev. Lett. 58:86–88 (1987).Google Scholar
  19. 19.
    U. Wolff, Collective Monte Carlo updating for spin systems.Phys. Rev. Lett. 62:361–364 (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • A. Brandt
    • 1
  • M. Galun
    • 1
  • D. Ron
    • 1
  1. 1.Department of Applied Mathematics and Computer ScienceWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations