Abstract
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(ε-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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
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].
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.
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.
A. Brandt, Multigrid methods in lattice field computations,Nucl. Phys. B. (Proc. Suppl.) 26:137–180 (1992).
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.
M. Galun, Optimal multigrid algorithms for model problems in statistical mechanics, M.Sc. Thesis, Weizmann Institute of Science (1992).
J. Goodman and A. D. Sokal, Multigrid Monte Carlo methods for lattice field theories,Phys. Rev. Lett. 56:1015–1018 (1986).
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).
D. Kandel and E. Domany, General cluster Monte Carlo dynamics,Phys. Rev. B. 43:8539 (1991).
D. Kandel, E. Domany, and A. Brandt, Simulations without critical slowing down—Ising and 3-state Potts models.Phys. Rev. B 40:330 (1989).
D. Kandel, E. Domany, D. Ron, A. Brandt, and E. Loh, Jr., Simulations without critical slowing down,Phys. Rev. Lett. 60:1591 (1988).
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).
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.
D. Ron, Development of fast numerical solvers for problems in optimization and statistical mechanics, Ph.D. Thesis, Weizmann Institute of Science (1989).
S. Shmulyian, Multilevel Monte Carlo algorithms for spin models, M.Sc. Thesis, Weizmann Institute of Science (1993).
A. D. Sokal, How to beat critical slowing-down: 1990 update,Nucl. Phys. B (Proc. Suppl.) 20:55–67 (1991).
R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations,Phys. Rev. Lett. 58:86–88 (1987).
U. Wolff, Collective Monte Carlo updating for spin systems.Phys. Rev. Lett. 62:361–364 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brandt, A., Galun, M. & Ron, D. Optimal multigrid algorithms for calculating thermodynamic limits. J Stat Phys 74, 313–348 (1994). https://doi.org/10.1007/BF02186816
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02186816