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Applications of Extended Ensemble Monte Carlo

Applications of Extended Ensemble Monte Carlo. Yukito IBA The Institute of Statistical Mathematics, Tokyo, Japan. Extended Ensemble MCMC. A Generic Name which indicates : Parallel Tempering, Simulated Tempering , Multicanonical Sampling , Wang-Landau , … Umbrella Sampling.

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Applications of Extended Ensemble Monte Carlo

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  1. Applications of Extended Ensemble Monte Carlo Yukito IBA The Institute of Statistical Mathematics, Tokyo, Japan

  2. Extended Ensemble MCMC A Generic Name which indicates: Parallel Tempering, Simulated Tempering, Multicanonical Sampling, Wang-Landau, … Umbrella Sampling Valleau and Torrie 1970s

  3. Contents 1. Basic Algorithms Parallel Tempering .vs Multicanonical 2. Exact Calculation with soft Constraints Lattice Protein / Counting Tables 3. Rare Events and Large Deviations Communication Channels Chaotic Dynamical Systems

  4. Basic Algorithms Parallel Tempering Multicanonical Monte Carlo

  5. References in physics • Iba (2001) Extended Ensemble Monte Carlo Int. J. Mod. Phys. C12 p.623. A draft version will be found at http://arxiv.org/abs/cond-mat/0012323 • Landau and Binder (2005) A Guide to Monte Carlo Simulations in Statistical Physics (2nd ed. , Cambridge) • A number of preprints will be found in Los Alamos Arxiv on the web. # This slide is added after the talk

  6. × × × Slow mixing by multimodal dist.

  7. Bridging fast mixing high temperature slow mixing low temperature

  8. Path Sampling 1.Facilitate Mixing 2.Calculate Normalizing Constant (“free energy”) “Path Sampling” Gelman and Meng (1998) stress 2. but 1. is also important In Physics: from 2. to 1. 1970s 1990s

  9. Parallel Tempering a.k.a. Replica Exchange MC Metropolis Coupled MCMC Geyer (1991), Kimura and Taki (1991) Hukushima and Nemoto (1996) Iba(1993, in Japanese) Simulate Many “Replica”s in Parallel MCMC in a Product Space

  10. Examples Gibbs Distributions with different temperatures Any Family parameterized by a hyperparameter

  11. Exchange of Replicas K=4

  12. Accept/Reject Exchange Calculate Metropolis Ratio Generate a Uniform Random Number in [0,1) and accept exchange iff

  13. Detailed Balance in Extended Space Combined Distribution

  14. Multicanonical Monte Carlo Berg et al. (1991,1992) sufficient statistics Energy not Expectation Exponential Family sufficient statistics

  15. Density of States The number of which satisfy

  16. Multicanonical Sampling

  17. Weight and Marginal DistributionOriginal (Gibbs) Multicanonical Random

  18. flat marginal distribution Scanning broad range of E

  19. Reweighting Formally, for arbitraryit holds. Practically, is required, else the variance diverges in a large system.

  20. Q.How can we do without knowledge onD(E) Ans． Estimate D(E) in the preliminary runs kth simulation Simplest Method : Entropic Sampling in

  21. Estimation of Density of States (Ising Model on a random net) 30000 MCS 2 k=1 3 4 5 10 11 14 k=15

  22. Estimation of D(E) Entropic Sampling • Histogram • Piecewise Linear • Fitting, Kernel Density Estimation .. • Wang-Landau • Flat Histogram Original Multicanonical Continuous Cases D(E)dE : Non-trivial Task

  23. Parallel Tempering / Multicanonical parallel tempering combined distribution simulated tempering mixture distribution to approximate

  24. Potts model (2-dim, q=10 states) disordered ordered

  25. Phase Coexistence/ 1st order transition parameter (Inverse Temperature) changes sufficient statistics (Energy) jumps water and ice coexists

  26. bridging by multicanoncal construction Potts model (2-dim, q=10 states) disordered ordered

  27. Comparison @ Simple Liquids , Potts Models .. Multicanonical seems better than Parallel Tempering @But, for more difficult cases ? ex. Ising Model with three spin Interaction

  28. Soft Constraints Lattice Protein Counting Tables The results on Lattice Protein are taken from joint works with G Chikenji (Nagoya Univ) and Macoto Kikuchi (Osaka Univ) Some examples are also taken from the other works by Kikuchi and coworkers.

  29. Lattice Protein Model Motivation Simplest Models of Protein Lattice Protein : Prototype of “Protein-like molecules” Ising Model : Prototype of “Magnets”

  30. Lattice Protein (2-dim HP)

  31. sequence of FIXED and corresponds to 2-types of amino acids (H and P) conformation of chain STOCHASTIC VARIABLE SELF AVOIDING (SELF OVERLAP is not allowed) IMPORTANT!

  32. Energy (HP model) the energy of conformationx is defined as E(X)= - the number of in x

  33. Examples E= -1 E=0 Here we do not count the pairs neighboring on the chain but it is not essential because the difference is const.

  34. MCMC Slow Mixing Even Non-Ergodicity with local moves Bastolla et al. (1998) Proteins 32 pp. 52-66 Chikenji et al. (1999) Phys. Rev. Lett. 83 pp.1886-1889

  35. Multicanonical Multicanonical w.r.t. E only NOT SUFFUCIENT Self-Avoiding condition is essential

  36. Soft Constraint Self-Avoiding condition is essential Soft Constraint is the number of monomers that occupy the site i

  37. Samples with are used for the calculation of the averages Multi Self-Overlap Sampling Multi Self-Overlap Ensemble Bivariate Density of States in the (E,V) plane V (self-overlap) E EXACT !!

  38. Generation of Paths by softening of constraints E V=0 large V

  39. Comparison with multicanonical with hard self-avoiding constraint switching between three groups of minimum energy states of a sequence conventional (hard constraint) proposed (soft constraint)

  40. optimization

  41. optimization (polymer pairs) Nakanishi and Kikuchi (2006) J.Phys.Soc.Jpn. 75 pp.064803 / q-bio/0603024

  42. double peaks An Advantage of the method is that it can use for the sampling at any temperature as well as optimization 3-dim Yue and Dill (1995) Proc. Nat. Acad. Sci. 92 pp.146-150

  43. non monotonic change of the structure Another Sequence Chikenji and Kikuchi (2000) Proc. Nat. Acad. Sci 97 pp.14273 - 14277

  44. Related Works Self-Avoiding Walk without interaction / Univariate Extension Vorontsov-Velyaminov et al. : J.Phys.Chem.,100,1153-1158 (1996) Lattice Protein but not exact / Soft-Constraint without control Shakhnovich et al. Physical Review Letters 67 1665 (1991) Continuous homopolymer -- Relax “core” Liu and Berne J Chem Phys 99 6071 (1993) See References in Extended Ensemble Monte Carlo, Int J Phys C 12 623-656 (2001) but esp. for continuous cases, there seems more in these five years

  45. Counting Tables Pinn et al. (1998) Counting Magic Squares Soft Constraints + Parallel Tempering

  46. Multiple Maxima Parallel Tempering Sampling by MCMC

  47. Normalization Constant calculated by Path sampling (thermodynamic integration)

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