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Multi-level approaches to discrete-state and stochastic problems

  • Achi Brandt
  • Dorit Ron
  • Daniel J. Amit
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)

Abstract

Fast multi-level techniques are developed for large-scale problems whose variables may assume only discrete values (such as spins with only “up” and “down” states), and/or where the relations between variables is probabilistic. Motivation and examples are taken from statistical mechanics and field theory. Detailed procedures are developed for the fast global minimization of discretestate functionals, or other functionals with many local minima, using new principles of multilevel interactions. Tests with Ising spin models are reported. Of special interest to physicists are the Ising model in a random field and spin glasses, which are known to lead to difficulties in conventional Monte-Carlo algorithms.

Keywords

Ising Model Coarse Level Single Spin Attraction Basin Spin Glass Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    D. Andelman, H. Orland and L.C.R. Wijewardhana: Lower critical dimension of the random-field Ising model: a montecarlo study. Phys. Rev. Lett. 52 (1984), 145.CrossRefGoogle Scholar
  2. [2]
    J.C. Angles D'Auriac and R. Maynard: On the random antiphase state in the ± J spin glass model in two dimensions. Solid State Commun. 49 (1984), 785.CrossRefGoogle Scholar
  3. [3]
    J.C. Angles D'Auriac, M. Preissmann and R. Rammal: The random field Ising model: algorithmic complexity and phase transition. J. Physique Lett. 46 (1985), 173.CrossRefGoogle Scholar
  4. [4]
    J.C. Angles D'Auriac and R. Rammal: Phase Diagram of the Random Field Ising Model. CNR Grenoble, preprint, 1986.Google Scholar
  5. [5]
    F. Barahona: On the computational complexity of Ising spin glass models. J. Phys. A15 (1982), 3241.MathSciNetGoogle Scholar
  6. [6]
    F. Barahona, R. Maynard, R. Rammal and J.P. Uhry: Morphology of ground states of the two-dimensional frustration model. J. Phys. A15 (1982), 673.MathSciNetGoogle Scholar
  7. [7]
    G. Bhanot, M. Creutz and H. Neuberger: Microcanonical simulation of Ising systems. Nucl. Phys. FS B235 (1984), 417.CrossRefGoogle Scholar
  8. [8]
    I. Bieche, R. Maynard, R. Rammal and J.P. Uhry: On the ground states of the frustration model of a spin glass by a matching method in graph theory. J. Phys. A13 (1980), 2553.MathSciNetGoogle Scholar
  9. [9]
    E. Bonomi and J-L Lutton: The N-city travelling salesman problem: statistical mechanics and the metropolis algorithm. SIAM Review 26 (1984), 551. See also [16], [17] and [18].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Brandt: Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. Monograph, 191 pages. Available as GMD Studien 85 from GMD-AIW, Postfach 1240, D-5205, St. Augustin 1, W. Germany.Google Scholar
  11. [11]
    M. Falcioni, E. Marinari, M.L. Paciello, G. Parisi, B. Taglienti and Y.S. Zhang: Large distance correlation functions for an SU(2) lattice gauge theory. Nucl. Phys. FS B215 (1983), 265.MathSciNetCrossRefGoogle Scholar
  12. [12]
    K.H. Fisher: Spin glasses I. Phys. Stat. Solidi B116 (1983), 357.CrossRefGoogle Scholar
  13. [13]
    K.H. Fisher: Spin glasses II. Phys. Stat. Solidi B130 (1985), 13.CrossRefGoogle Scholar
  14. [14]
    R. Graham and F. Haake: Quantum Statistics in Optics and Solid-State Physics. Springer Tracts in Modern Physics, Vol. 66, Springer-Verlag, Berlin, 1973.Google Scholar
  15. [15]
    A. Hasenfratz and P. Hasenfratz: Lattice Gauge Theories. Florida State University, preprint FSU-SCRI 85-2, 1985.Google Scholar
  16. [16]
    S. Kirkpatrick: Models of disordered systems. Lecture Notes in Physics 149 (C. Castellani et al., eds.), Springer-Verlag, Berlin, 280.Google Scholar
  17. [17]
    S. Kirkpatrick, C.D. Gelatt Jr. and M.P. Vecchi: Optimization by simulated annealing. Science 220 (1983), 671.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S. Kirkpatrick and G. Toulouse: Configuration space analysis of travelling salesman problems. J. Physique 46 (1985), 1277.MathSciNetCrossRefGoogle Scholar
  19. [19]
    S.K. Ma: Renormalization group by Monte-Carlo methods. Phys. Rev. Lett. 37 (1976), 461.CrossRefGoogle Scholar
  20. [20]
    N.D. Mackenzie and A.P. Young: Lack of ergodicity in the infinite range Ising spin-glass. Phys. Rev. Lett. 49 (1982), 301.CrossRefGoogle Scholar
  21. [21]
    N.D. Mackenzie and A.P. Young: Statics and dynamics of the infinite range Ising spin glass model. J. Phys. C16 (1983), 5321.Google Scholar
  22. [22]
    E. Marinari, G. Parisi and C. Rebbi: Computer estimates of meson masses in SU(2) lattice gauge theory. Phys. Rev. Lett. 47 (1981), 1795. For a recent review see [15].CrossRefGoogle Scholar
  23. [23]
    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. 21 (1953), 1087.CrossRefGoogle Scholar
  24. [24]
    A.T. Ogielski and I. Morgenstern: Critical behavior of three-dimensional Ising model of spin glass. J. Appl. Phys. 57 (1985), 3382.CrossRefGoogle Scholar
  25. [25]
    G. Parisi, R. Petronzio and F. Rapuano: A measurement of the string tension near the continuum limit. Phys. Lett. 128B (1983), 418.CrossRefGoogle Scholar
  26. [26]
    G. Parisi and Y.S. Wu: Perturbation theory without gauge fixing. Scientia Sinica 24 (1981), 483.MathSciNetGoogle Scholar
  27. [27]
    H. Sompolinsky: Time dependent order parameters of spin-glasses. Phys. Rev. Lett. 47 (1981), 935.CrossRefGoogle Scholar
  28. [28]
    H. Sompolinsky and A. Zippelius: Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses. Phys. Rev. B25 (1982), 6860.CrossRefGoogle Scholar
  29. [29]
    R.L. Stratonovich: Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, New York, 1963.Google Scholar
  30. [30]
    R.H. Swendsen: Monte-Carlo renormalization group. Phys. Rev. Lett. 42 (1979), 859.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Achi Brandt
    • 1
  • Dorit Ron
    • 1
  • Daniel J. Amit
    • 2
  1. 1.Department of Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Racah Institute of PhysicsThe Hebrew UniversityJerusalemIsrael

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