Parallel cluster labeling for large-scale Monte Carlo simulations

We present an optimized version of a cluster labeling algorithm previously introduced by the authors. This algorithm is well suited for large-scale Monte Carlo simulations of spin models using cluster dynamics on parallel computers with large numbers of processors. The algorithm divides physical space into rectangular cells which are assigned to processors and combines a serial local labeling procedure with a relaxation process across nearest-neighbor processors. By controlling overhead and reducing inter-processor communication this method attains good computational speed-up and efficiency. Large systems of up to 655362 spins have been simulated at updating speeds of 11 nanosecs/site (90.7 × 106 spin updates/sec) using state-of-the-art supercomputers. In the second part of the article we use the cluster algorithm to study the relaxation of magnetization and energy on large Ising models using Swendsen-Wang dynamics. We found evidence that exponential and power law factors are present in the relaxation process as has been proposed by Hackl et al. The variation of the power-law exponent λM taken at face value indicates that the value of ZM falls in the interval 0.31–0.49 for the time interval analysed and appears to vanish asymptotically.

[1]  Claudio Rebbi,et al.  Lattice Gauge Theories And Monte Carlo Simulations , 1983 .

[2]  Uzi Vishkin,et al.  An optimal parallel connectivity algorithm , 1984, Discret. Appl. Math..

[3]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[4]  Frithjof Karsch,et al.  Large Scale Computational Physics on MAssively Parallel Computers , 1993 .

[5]  Anthony N. Burkitt,et al.  Parallel algorithms in computational science , 1991, Springer series in information sciences.

[6]  Gupta,et al.  Monte Carlo renormalization-group study of the three-dimensional Ising model. , 1992, Physical review. B, Condensed matter.

[7]  Dynamics of Ising ferromagnets in droplet model , 1991 .

[8]  Clive F. Baillie,et al.  Cluster Identification Algorithms for Spin Models - Sequential and Parallel , 1991, Concurr. Pract. Exp..

[9]  Uzi Vishkin,et al.  An O(log n) Parallel Connectivity Algorithm , 1982, J. Algorithms.

[10]  Coddington,et al.  Comparison of cluster algorithms for two-dimensional Potts models. , 1991, Physical review. B, Condensed matter.

[11]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[12]  E. N. Miranda.  Swendsen-Wang cluster relaxation time in the 3D ising model , 1991 .

[13]  K. Binder.  Monte Carlo methods in statistical physics , 1979 .

[14]  Guy E. Blelloch,et al.  Vector Models for Data-Parallel Computing , 1990 .

[15]  Scaling Ansatz for Swendsen-Wang dynamics. , 1989, Physical review letters.

[16]  Alan M. Ferrenberg,et al.  Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study. , 1991, Physical review. B, Condensed matter.

[17]  J. M. Singer,et al.  3D Ising model with Swendsen-Wang dynamics: a parallel approach , 1994 .

[18]  Coddington,et al.  Empirical relations between static and dynamic exponents for Ising model cluster algorithms. , 1992, Physical review letters.

[19]  Pablo Tamayo.  Magnetization relaxation to equilibrium on large 2D Swendsen-Wang Ising models , 1993 .

[20]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[21]  R. Brower,et al.  Single-cluster Monte Carlo dynamics for the Ising model , 1990 .

[22]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[23]  Dietrich Stauffer,et al.  From Newton to Mandelbrot : a primer in theoretical physics with fractals for the Macintosh , 1991 .

[24]  John Greiner,et al.  A comparison of parallel algorithms for connected components , 1994, SPAA '94.

[25]  Wolff.  Lattice field theory as a percolation process. , 1988, Physical review letters.

[26]  R. Hackl,et al.  PARALLELIZATION OF THE 2D SWENDSEN–WANG ALGORITHM , 1993 .

[27]  David A. Bader,et al.  Parallel Algorithms for Image Histogramming and Connected Components with an Experimental Study , 1996, J. Parallel Distributed Comput..

[28]  J. M. Singer,et al.  Efficient parallelization of the 2D Swendsen-Wang algorithm , 1994 .

[29]  Huang,et al.  Random surface dynamics for Z2 gauge theory. , 1990, Physical review. D, Particles and fields.

[30]  Dynamics of Swendsen-Wang clusters in 2D Ising model , 1991 .

[31]  Coniglio.  Fractal structure of Ising and Potts clusters: Exact results. , 1989, Physical review letters.

[32]  Hidetoshi Mino,et al.  A vectorized algorithm for cluster formation in the Swendsen-Wang dynamics , 1991 .

[33]  E. N. Miranda.  Geometrical properties of Swendsen-Wang clusters , 1991 .

[34]  Kurt Binder,et al.  Monte Carlo Simulation in Statistical Physics , 1992, Graduate Texts in Physics.

[35]  Andrew I. Mel'cuk,et al.  Relaxation to equilibrium of single cluster dynamics of the Ising model , 1994 .

[36]  R. Brower,et al.  A parallel multigrid algorithm for percolation clusters , 1991 .

[37]  Antonio Coniglio,et al.  Clusters and Ising critical droplets: a renormalisation group approach , 1980 .

[38]  Barkema,et al.  Parallel simulation of the Ising model. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  U. Wolff.  Comparison Between Cluster Monte Carlo Algorithms in the Ising Model , 1989 .

[40]  Jorge L. C. Sanz,et al.  Hypercube and Shuffle-Exchange Algorithms for Image Component Labeling , 1987, J. Algorithms.

[41]  Enzo Marinari,et al.  New Simd Algorithms For Cluster Labeling On Parallel Computers , 1993 .

[42]  Raoul Kopelman,et al.  Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm , 1976 .

[43]  E. N. Miranda.  Droplet dynamics in the 3D Ising model , 1991 .