Abstract
We introduce the Markov chain Monte Carlo method and review the background of the cluster algorithms in statistical physics. One of the first such successful algorithm was developed by Swendsen and Wang eight years ago. In contrast to the local algorithms, cluster algorithms update dynamical variables in a global fashion. Therefore, large changes are made in a single step. The method is very efficient, especially near the critical point of a second-order phase transition. Studies of various statistical mechanics models and some generalizations of the algorithm will be briefly reviewed. We mention applications in other fields, especially in imaging processing.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., Vol. 21, pp. 1087–1092, 1953.
M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Volume I: Basics, John Wiley & Sons, New York, 1986.
L. E. Reichl, A modern Course in Statistical Physics, Edward Arnold, 1980.
E. Ising, “Beitrag zur theorie des ferromagnetismus,” Z. Phys., Vol. 31, pp. 253, 1925.
L. Onsager, “Crystal statistics I: A two-dimensional model with an order-disorder transition,” Phys. Rev., Vol. 65, pp. 117–149, 1944.
For reviews, see A. D. Sokal, “Monte Carlo methods in statistical mechanics: foundations and new algorithms,” Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne, 1989; J.-S. Wang and R. H. Swendsen, “Cluster Monte Carlo algorithms,” Physica A, Vol. 167, pp. 565–579, 1990; R. H. Swendsen, J.-S. Wang, and A. M. Ferrenberg, “New Monte Carlo methods for improved efficiency of computer simulation in statistical mechanics,” in The Monte Carlo Method in Condensed Matter Physics, ed. K. Binder, Topics in Applied Physics, Vol. 71, pp. 75–91, Springer, Berlin, 1992.
R. H. Swendsen and J.-S. Wang, “Non-universal critical dynamics in Monte Carlo simulations,” Phys. Rev. Lett., Vol. 58, pp. 86–88, 1987.
D. Stauffer and A. Aharony, “Introduction to Percolation Theory,” 2nd ed, Taylor & Francis, London, 1992.
C. M. Fortuin and P. W. Kasteleyn, “On the random cluster model. I. Introduction and relation to other models,” Physica (Utrecht) Vol. 57, pp. 536–564, 1972.
U. Wolff, “Collective Monte Carlo updating for spin systems,” Phys. Rev. Lett., Vol. 62, pp. 361–364, 1989.
R. Brower, “Embedded dynamics for φ 4 theory,” Phys. Rev. Lett., Vol. 62, pp. 1087–1090, 1989.
J.-S. Wang, R. H. Swendsen, and R. Kotecký, “Antiferromagnetic Potts models,” Phys. Rev. Lett. Vol. 63, pp. 109–112, 1989.
D. Kandel, E. Domany, D. Ron, and A Brandt, “Simulations without critical slowing down”, Phys. Rev. Lett., Vol. 60, pp. 1591–1594, 1988.
D. Kandel, E. Domany, and A. Brandt, “Simulations without critical slowing down: Ising and three-state Potts models,” Phys. Rev. B, vol. 40, pp. 330–344, 1989.
D. Kandel, R. Ben-Av, and E. Domany, “Cluster dynamics for fully frustrated systems,” Phys. Rev. Lett, Vol. 65, pp. 941–944, 1990.
S. Liang, “Application of cluster algorithms to spin glasses,” Phys. Rev. Lett., Vol. 69, pp. 2145–2148, 1992.
H. G. Evertz, G. Lana, and M. Marcu, “Cluster algorithms for vertex models,” Phys. Rev. Lett., Vol. 70, pp. 875–879, 1993.
R. G. Edwards and A. D. Sokal, “Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm,” Phys. Rev. D, Vol. 38, pp. 2009–2012, 1988.
J. Besag and P. J. Green, “Spatial statistics and Bayesian computation,” J. R. Statist. Soc. B, Vol. 55, pp. 25–37 and 53–102, 1993.
A. J. Gray, J. W. Kay, and D. M. Titterington, “An empirical study of the simulation of various models used for images,” IEEE Trans. Pattern Anal. Machine Intell., Vol. 16, pp. 507–513, 1994.
G. G. Potamianos and J. K. Goutsias, “Partition function estimation of Gibbs random field images using Monte Carlo simulations,” IEEE Trans. Inform. Theory, Vol. 39, pp. 1322–1332, 1993.
A. J. Gray, “Simulating posterior Gibbs distributions: a comparison of the Swendsen-Wang and Gibbs sampler methods,” Statistics and Computing, Vol. 4, pp. 189–201, 1994.
I. Gaudron, “Vitesse de convergence de la dynamique de Swendsen-Wang pour minimiser des énergies de segmentation d'images,” C. R. Acad. Sci. Paris, t. 320, Série I, pp. 475, 1995.
I. Gaudron, “Rate of convergence of the Swendsen-Wang dynamics in image segmentation problems: A theoretical and experimental study,” J. Stat. Phys., 1996.
M. A. Novotny, “Monte Carlo algorithms with absorbing Markov Chains: Fast local algorithms for slow dynamics,” Phys. Rev. Lett., Vol. 74, pp. 1–5, 1995.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wang, JS. (1996). Cluster Monte Carlo algorithms and their applications. In: Li, S.Z., Mital, D.P., Teoh, E.K., Wang, H. (eds) Recent Developments in Computer Vision. ACCV 1995. Lecture Notes in Computer Science, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60793-5_85
Download citation
DOI: https://doi.org/10.1007/3-540-60793-5_85
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60793-9
Online ISBN: 978-3-540-49448-5
eBook Packages: Springer Book Archive