Monte Carlo Methods

NIHES EERC Oct. 25, 2002

Monte Carlo MethodsMark A. Novotny

Dept. of Physics and Astronomy

Mississippi State U.

[email protected]

Supported in part by NSF DMR0120310 and ITR/AP(MPS)0113049

Random topics about random things for random people


Where is Monte Carlo?

• Europe• Principality of Monaco• Monte Carlo is 1 of 5

regions of Monaco• Monte Carlo founded in

1866 by Prince Charles III• Renowned casino,

luxurious hotels, beaches, …


What is a Monte Carlo simulation?

•In a Monte Carlo simulation we attempt to follow the `time dependence’ of a model for which change, or growth, does not proceed in some rigorously predefined fashion (e.g. according to Newton’s equations of motion) but rather in a stochastic manner which depends on a sequence of random numbers which is generated during the simulation.•Landau and Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge U. Press, Cambridge U.K., 2000), p. 1.


Birth of the Monte Carlo method

• Los Alamos National Laboratory• 1953 Physical Review• Metropolis, Rosenbluth,

Rosenbluth, Teller, and Teller• Conferences/MonteCarloMethods/

“The only good Monte Carlo is a dead Monte Carlo.” Trotter and Turkey, 1954

http://cnls.lanl.gov/Conferences/MonteCarloMethods/


OUTLINE

• Random numbers• Integrals via Monte

Carlo• Importance Sampling

Monte Carlo• Dynamic Monte Carlo• Decay of nuclei (birth-

death processes)

• Random motion in 1 dimension

• Long-time simulations (Monte Carlo with Absorbing Markov Chains, MCAMC)

• Parallel dynamic Monte Carlo simulations


OUTLINE




death processes)





(not-really, but almost) Random Numbers

• Uniformly distributed numbers in [0,1]• How good is `good enough’?• `religious question’• Linear congruential, R250R250, Marsaglia, 4-tap4-tap,

system supplied

•Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.•John von Neumann (1951)


(not-really, but almost) Random Numbers from R250

• Xn = XOR( Xn-p , Xn-q )• XOR(,) is exclusive OR operator• With p2+q2+1=prime • (p and q are Mersine primes)

• R98: p=98, q=27• R250: p=250, q=103• R1279: p=1279, q=216 or 418• R9689: p=9689, q=84, 471, 1836, 2444, or 4187


OUTLINE




death processes)





Calculate =pi using Monte Carlo

• Set Nin=0• Do N times

o Calculate 3 random numbers, r1, r2, r3

o Let x=r1

o Let y=r2

o Use r3 to choose quadrant (change signs of x and y)

o If x2+y2.le.1 set Nin=Nin+1

• Estimate for =pi = 4 Nin / N

Calculate using Monte Carlo

N=103 N=104 N=105

Live or die by the Law of Large Numbers


OUTLINE




death processes)





Importance Sampling Monte Carlo

• Old way: choose points with equal probability from rectangle with a<xi<b and

0<y<y0; and then use estimate yestimate=i f(xi) (b-a) y0

• Rather choose points with importance of the value of the function at that point to the integral, p(x)

• Estimate of integral yestimate=i p-1(xi) f(xi)

a b

y0

0


OUTLINE




death processes)





Dynamic Monte Carlo

• For Static Monte Carlo the order of generation of points does not matter (like finding the integrals)

• For Dynamic Monte Carlo the order does matter• This gives a Markov chain method, governed by the Master

Equation


OUTLINE




death processes)





Birth and Death processes

• Decay (death) of nuclei (organisms)• rate =0.001 /msec• N0=100

• N(t) = N0 exp(- t)

• t1/2 = ln(2)/= 693.15


Birth and Death processes• Start with N=N0 individuals• Start time t=0• Assume that decay (death) constant <1• For each nucleus which has not yet decayed

Generate a random number 0<r<1 Nucleus decays if r<, and set N=N-1, else the

nucleus remains• When done sampling all remaining nuclei, set

t=t+1• Repeat until all nuclei have decayed


Birth and Death processesEvent Driven --- n-fold way

• Start with N=N0 individuals• Start time t=0• Assume that decay (death) constant <1• For each nucleus which has not yet decayed

Generate a random number 0<r<1 Nucleus decays if r<, and set N=N-1, else the nucleus

remains• When done sampling all remaining nuclei, set t=t+1• Repeat until all nuclei have decayed

May be VERY SLOW if decay rate is small!!!

Does the nucleus decay? No no no no no no no no no no no no no no no ...


Birth and Death processesEvent Driven -- or -- n-fold way

• Start with N=N0 individuals• Start time t=0• Assume that decay (death) constant <1• Until all the nuclei have decayed do:

Generate two random number 0<r1 ,r2<1 Calculate time to leave current state: t=-ln(r1)/(N ) Set t = t + t Set N = N - 1 Use r2 to pick which of remaining nuclei decayed

One decay every algorithmic step, no matter how small is


Birth and Death processes

• Birth and death of organisms• grow=0.001 /msec

die =0.001 /msec

• N0=100


OUTLINE




death processes)





Random Walker coagulation

Albena, Bulgaria model (2002)



Albena, Bulgaria model (2002)





Event driven (n-fold way)

One algorithmic step

step from current configuration



Monte Carlo with Absorbing Markov Chains (MCAMC) (s=2)

In one algorithmic step

step out of low valleys



Average time until coagulation

All algorithms statistically the same



Different algorithms

require different amounts of computer time


Random motion --- ion channel flow

Random motion (of a random walker) through a channel











Average lifetimes the same for all algorithms

Computer times different for different algorithms



Square lattice for random walker, lifetime is time to get from one end to another


OUTLINE




death processes)





Monte Carlo for Ising models



Use spin classes for Ising model --- square lattice has 10 spin classes

•Every spin belongs to one of 10 classes

•Probability of flipping a spin in each class is the same

•If chosen, probability of flipping spin in class i is p(i)

•Number of spins in class i is ci



A GAME



MCAMC

n-fold way



n-fold way

with needed Bookkeeping



MCAMC



Square lattice Ising model



MCAMC

resultssquare Ising



Age of universe in femtoseconds?

10-15 seconds


OUTLINE




death processes)





Dynamic Monte CarloParallel Discrete Event Simulations

• Dynamics for Materials• Dynamics for biological

and ecological models• Dynamics of Magnets• Cell-phone switching• Spread of infectious

diseases• Resource allocation

following terrorist attacks• War-game scenarios


“Nature”?

computers & algorithms

Parallel Discrete Event Simulations

Best is trivial parallelizationEach processor runs same program with

different random number sequence


Two approaches to parallelization

ConservativePE “idles” if causality is not guaranteedutilization, u: fraction of non-idling PEs

i

(site index) i

d=1

Optimistic (or speculative)PEs assume no causality violationsRollbacks to previous states once causality violation is found (extensive state saving or reverse simulation)Rollbacks can cascade (“avalanches”)

d=2


Non-equilibrium surface growthe.g., kinetic roughening as in Molecular Beam Epitaxy

2

1

2 )]()([1)( ttL

twdL

iid

dL

iid t

Lt

1

)(1)( Surface width = w

Surface is FRACTAL


Utilization/efficiencyFinite-size effects for the density of local minima/average growth rate (steady state):

2464.0 u

Lconstuu L

. (d=1)


Implications for scalabilitySimulation reaches steady state for long times

Simulation phase: scalable

Measurement (data management) phase: not scalable

)1(2

.

Lconstuu L

u asymptotic average growth rate (simulation speed or utilization) is non-zero

22 ~ Lw L

wmeasurement at meas:


Actual implementation

1. Local time incremented 2. If chosen site is on the boundary, PE must wait until min{nn}

Dynamics of a thin magnetic film


The Physics of Queuing

• PDES applicable to many situations

• Can be made scalable (patent applied for)

• Use ideas from physics to understand & improve computer science & biological applications


References and LinksNovotny and coworkers

• For a review of advanced dynamic methods see: M.A. Novotny in Annual Reviews of Computational Physics IX, ed. D. Stauffer, (World Scientific, Singapore, 2001), p. 153; preprint xxx.lanl.gov/cond-mat/0109182 click---HERE.

• Metastability: Physical Review Letters, 81, 834 (1998).• Monte Carlo with Absorbing Markov Chains: Physical Review Letters, 74, 1 (1995); erratum

75, 1424 (1995). • Discrete-time n-fold way: Computers in Physics, 9, 46 (1995).• Parallel Discrete Event Simulations, General: Physical Review Letters, 84, 1351 (2000).• Implementation of non-trivial parallel Monte Carlo: Journal of Computational Physics, 153, 488

(1999).• Projective Dynamics: Physical Review Letters, 80, 3384 (1998).• Constrained Transfer Matrix for metastability: Physical Review Letters, 71, 3898 (1993).

http://xxx.lanl.gov/abs/cond-mat/0109182

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Monte Carlo Methods

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