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Applications of Extended Ensemble Monte Carlo
Yukito IBA
The Institute of Statistical Mathematics, Tokyo, Japan
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Extended Ensemble MCMC
A Generic Name which indicates:
Parallel Tempering,
Simulated Tempering,
Multicanonical Sampling,
Wang-Landau, …
Umbrella Sampling Valleau and Torrie1970s
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Contents1. 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
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Basic Algorithms
Parallel Tempering
Multicanonical Monte Carlo
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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
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Slow mixing by multimodal dist.
××
××××
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Bridging
fast mixinghigh temperature
slow mixinglow temperature
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Path Sampling
1.Facilitate Mixing2.Calculate Normalizing Constant (“free energy”)
In Physics: from 2. to 1.1970s 1990s
“Path Sampling” Gelman and Meng (1998)
stress 2. but 1. is also important
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Parallel Tempering
a.k.a. Replica Exchange MC
Metropolis Coupled MCMC
Simulate Many “Replica”s in Parallel
MCMC in a Product Space
Geyer (1991), Kimura and Taki (1991)
Hukushima and Nemoto (1996)
Iba(1993, in Japanese)
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Examples
Gibbs Distributions with different temperatures
Any Family parameterized by
a hyperparameter
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Exchange of Replicas
K=4
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Accept/Reject Exchange
Calculate Metropolis Ratio
Generate a Uniform Random Number
in [0,1) and accept exchange
iff
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Detailed Balance in Extended Space
Combined Distribution
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Multicanonical Monte Carlo
sufficient statistics
sufficient statisticssufficient statistics
Exponential Family
Energy not Expectation
Berg et al. (1991,1992)
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Density of States
The number of which satisfy
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Multicanonical Sampling
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Weight and Marginal Distribution Original (Gibbs) Multicanonical Random
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flat marginal distribution
Scanning broad range of Scanning broad range of E E
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Reweighting
Formally, for arbitrary it holds.
Practically, is required,
else the variance diverges in a large system.
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Q. How can we do without knowledge on D(E)Ans. Estimate D(E) in the preliminary runs
k th simulation
Simplest Method : Entropic Sampling
in
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Estimation of Density of States
55
k=1k=1 22
44 1010
k=15k=1514141111
30000 MCS30000 MCS33
(Ising Model on a random net)(Ising Model on a random net)
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Estimation of D(E)
• Histogram
• Piecewise Linear
• Fitting, Kernel Density Estimation ..
• Wang-Landau
• Flat Histogram
Entropic Sampling
Original Multicanonical
Continuous Cases D(E)dE : Non-trivial Task
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Parallel Tempering / Multicanonical
parallel tempering combined distributionsimulated tempering mixture distribution
to approximate
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disorderedordered
Potts model (2-dim, q=10 states)
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Phase Coexistence/ 1st order transition
parameter (Inverse Temperature) changes
sufficient statistics (Energy) jumps
water and ice coexists
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disordereddisorderedorderedordered
Potts model (2-dim, q=10 states)
bridging by multicanoncal constructionbridging by multicanoncal construction
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Comparison
@ Simple Liquids , Potts Models .. Multicanonical seems better than Parallel Tempering
@ But, for more difficult cases ?
ex. Ising Model with three spin Interaction
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Soft Constraints
Lattice Protein
Counting Tables
The results on Lattice Protein are taken from joint workswith G Chikenji (Nagoya Univ) and Macoto Kikuchi (Osaka Univ)
Some examples are also taken from the other worksby Kikuchi and coworkers.
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Lattice Protein Model
Motivation
Simplest Models of Protein
Lattice Protein :
Prototype of “Protein-like molecules”
Ising Model :
Prototype of “Magnets”
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Lattice Protein (2-dim HP)
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FIXEDsequence of
conformation of chain STOCHASTIC VARIABLE
SELF AVOIDING(SELF OVERLAP is not allowed)
IMPORTANT!
and corresponds to 2-types of amino acids (H and P)
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E(X)= - the number of
Energy (HP model)
in x
the energy of conformation x is defined as
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Examples
Here we do not count the pairs neighboring on the chainbut it is not essential because the difference is const.
E=0
E= -1
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MCMC
Slow Mixing
Even Non-Ergodicity with local moves
Chikenji et al. (1999)Phys. Rev. Lett. 83 pp.1886-1889
Bastolla et al. (1998) Proteins 32 pp. 52-66
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Multicanonical
Multicanonical w.r.t. E only
NOT SUFFUCIENT
Self-Avoiding condition is essential
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Soft Constraint
Self-Avoiding condition is essential
Soft Constraint
is the number ofmonomers that occupy the site i
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Multi Self-Overlap Sampling
Multi Self-Overlap Ensemble
Bivariate Density of States
in the (E,V) plane
E
V (self-overlap)
Samples withare used for the calculationof the averages
EXACT !!
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V=0 large V
E
Generation of Paths by softening of constraints
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Comparison with multicanonical with hard self-avoiding constraint
conventional(hard constraint)
proposed(soft constraint)
switching between
three groups ofminimum energy
statesof a sequence
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optimization
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optimization (polymer pairs)
Nakanishi and Kikuchi (2006)J.Phys.Soc.Jpn. 75 pp.064803 / q-bio/0603024
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double peaks
An Advantage of the methodis that it can usefor the sampling at any temperature as well as optimization
3-dim
Yue and Dill (1995) Proc. Nat. Acad. Sci. 92 pp.146-150
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Another Sequence
non monotonic changeof the structure
Chikenji and Kikuchi (2000)Proc. Nat. Acad. Sci 97
pp.14273 - 14277
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Related WorksSelf-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
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Counting Tables
4 9 2
3 5 7
8 1 6
Pinn et al. (1998)Counting Magic Squares
Soft Constraints+
Parallel Tempering
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Sampling by MCMC
Multiple Maxima
Parallel Tempering
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Normalization Constant
calculated by Path sampling (thermodynamic integration)
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Latin square (3x3)
For For each column, any given number column, any given number appears once and only once once and only once
For each raw, any given For each raw, any given number appears once and only once once and only once
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Latin square (26x26)
# This sample is taken from the web.# This sample is taken from the web.
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Counting Latin Squares
• 6
• 10
• 11
410000 MCS x 27 replicas
510000 MCS x 49 replicas
510000 MCS x 49 replicas
other 3 trials
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Counting Tables
Soft Constraints + Extended Ensemble MC
“Quick and Dirty” ways of calculating the number of tables that satisfy given constraints.
It may not be optimal for a special case,
but no case-by-case tricks, no mathematics,
and no brain is required.
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Rare Events and Large Deviations
Communication Channels #1
Chaotic Dynamical Systems #2
# 1 Part of joint works with Koji Hukushima (Tokyo Univ).
# 2 Part of joint works with Tatsuo Yanagita (Hokkaido Univ). (The result shown here is mostly due to him )
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Applications of MCMCStatistical Physics (1953 ~ )
Statistical Inference (1970s,1980s, 1990~)
Solution to any problem on
sampling & counting
estimation of large deviation
generation of rare events
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Noisy Communication Channel
prior encoded & degraded
decode distance (bit errors)by Viterbi, loopy BP,
MCMC
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Distribution of Bit Errors
Kronecker delta
tails of the distribution is not easy to estimate
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Introduction of MCMC
Sampling noise in channels by the MCMC
Given an error-correcting code
Some patterns of noise are very harmful
difficult to correct
Some patterns of noise are safe
easy to correct
NOT sampling from the posterior
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Multicanonical Strategy
MCMC sampling of
Broad distribution of
Broad distribution of distance
and
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Multicanonical Sampling
MCMC Sampling and
with the weight
Estimated by the iteration of preliminary runsexactly what we want,
but can be ..
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flat marginal distribution
Scanning broad range of bit errors
Enable efficient calculation of the tails of the distribution
(large deviation)
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Example
Convolutional Code
Binary Symmetric Channel Fix the number of noise (flipped bits)
Viterbi decoding
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SimplificationIn this case
is independent of Set
Binary Symmetric Channel Fix the number of noise (flipped bits)
sum over the possible positions of the noise
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Simulation
the number of bit errors
difficult tocalculate by simple
sampling
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Correlated Channels
It will be useful for the study of error-correcting code in a correlated channel.
Without assuming models of correlation
in the channel we can sample relevant
correlation patterns.
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Rare events in Dynamical Systems
Deterministic ChaosDoll et al. (1994), Kurchan et al. (2005)
Sasa, Hayashi, Kawasaki .. (2005 ~)
(Mostly) Stochastic DynamicsChandler Group
Frenkel et al.
and more …
Transition Path Sampling
Stagger and Step Method Sweet, Nusse, and Yorke (2001)
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Sampling Initial Condition
Sampling initial condition of
Chaotic dynamical systemsRare Events
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Double Pendulum
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control and stop the pendulumone of the three positions
Unstable fixed points
energy dissipation (friction) is assumedi.e., no time reversal sym.
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T is max time
Definition of artificial “energy”
stop = zero velocity
stopping position
penalty tolong time
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Metropolis step
Integrate Equation of Motion Integrate Equation of Motion andand
Simulate TrajectorySimulate Trajectory
Perturb Initial StatePerturb Initial State Evaluate “Energy”Evaluate “Energy”
Reject or AcceptReject or Accept
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for given Tfor given Tfor given Tfor given T
Parallel Tempering
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An animation by Yanagita is shown in the talk, but might not be seen on the web.
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Summary
Extended Ensemble + Soft Constraint strategy gives simple solutions to a number of difficult problems
The use of MCMC should not be restricted to the standard ones in Physics and Bayesian Statistics.
To explore new applications of MCMC extended ensemble MC will play an essential role.
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END