ST325 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION NUMBER 247 SEPTEMBER 1949 VOLUME 44 THE MONTE CARLO METHOD NICHOLAS METROPOLIS AND S. ULAM LOS ALAMOS LABORATORY

The Monte Carlo Method..

SESSION 1Presenter…

Mohomed Abraj.

Introduction

Classical statistical physics is a well understood subject which poses, however, many difficult problems when a concrete solution for interacting systems is sought. In almost all non-trivial applications, analytical methods can only provide approximate answers. Numerical computer simulations are, therefore, an important complementary method on our way to a deeper understanding of complex physical systems such as (spin) glasses and disordered magnets or of biologically motivated problems such as protein folding

Cont…

Quantum statistical problems in condensed matter or the broad field of elementary particle physics and quantum gravity are other major applications which, after suitable mappings, also rely on classical simulation techniques. we shall confine ourselves to a survey of computer simulations based on Markov chain Monte Carlo methods which realize the importance sampling idea.

Method and Approach

We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics.

The method is, Essentially, a statistical approach to the study of differential equations ,or more generally, of integro-differential equations that occur in various branches of the natural sciences.

ALREADY in the nineteenth century a sharp distinction began to appear between two different mathematical methods of treating physical phenomena.

Problems involving only a few particles were studied in classical mechanics, through the study of systems of ordinary differential equations.

For the description of systems with very many particles, an entirely different technique was used, namely, the method of statistical mechanics. In this latter approach, one does not concent rate on the individual particles but studies the properties of sets of particles

PHYSICAL SCALES FOR DILUTE GASES

Mean Free Path

Collision

Collision

MolecularDiameter

System Size

Gradient Scale

Quantum scale Kinetic scale Hydrodynamic scale

T/T

DSMC is the dominant numerical algorithm at the

kinetic scale

DSMC applications are expanding to multi-scale problems

Divide the system into cells and generate particles in each cell according to desired density, fluid velocity, and temperature.

From density, determine number of particles in cell volume, N, either rounding to nearest integer or from Poisson distribution.

Assign each particle a position in the cell, either uniformly or from the linear distribution using the density gradient.

From fluid velocity and temperature, assign each particle a velocity from Maxwell-Boltzmann distribution P(v; {u,T}) or from the Chapman-Enskog distribution .}),,,{;( TTP uuv

The Hamiltonian Function that is commonly used representing the energy of the model when using Monte Carlo Methods

we would see the great importance of Monte Carlo methods applied in Physics. Furthermore,

Monte Carlo methods also play significant role in quantum dynamics, physical chemistry, and related applied fields.

In quantum dynamics, Quantum Monte Carlo methods solve the multi-body problems for quantum system. In experimental particle physics.

Monte Carlo Methods are use for designing detectors, understanding their behavior and comparing experimental data to theory.

SESSION 2

Thank you

Presenter

Vishma Subodani

Particle Markov chain Monte Carlo methods

Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently.

CHARACTERISTICS OF MARKOV CHAIN

Irreducible ChainAperiodic ChainStationary Distribution

Markov Chain can gradually forget its initial state eventually converge to a unique stationary distribution

invariant distributionErgodic average

n

mttXf

mnf

1

)(1

TARGET DISTRIBUTION Target Distribution Function

(x)=ce-h(x)

h(x)in physics, the potential functionother system, the score function

cnormalization constant

make sure the integral of (x) is 1Presumably, all pdfs can be written in this form

Each technique has it advantages and disadvantages ,Broadly, for complex systems that may be subject to change later, the Monte-Carlo method is preferred because of its flexibility. For simpler systems, or studies to get a ‘feel’ for a problem, analytical methods may suffice

The decision as to whether the modeller should use analytical (e.g. deterministic equations) or simulation (i.e. Monte-Carlo) methods may be influenced by the following factors:

Complexity Scope Accuracy. Future development Application

Comparison of method and Accuracy of results

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SOME ADVANTAGES OF MC

Often the only type of model possible for complex systemsAnalytical models frequently infeasible

Process of building simulation can clarify understanding of real system

Sometimes more useful than actual application of final simulation

Allows for sensitivity analysis and optimization of real system without need to operate real systemCan maintain better control over experimental conditions than real systemTime compression/expansion: Can evaluate system on slower or faster time scale than real system

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SOME DISADVANTAGES OF MC

May be very expensive and time consuming to build simulation Easy to misuse simulation by “stretching” it beyond the limits of credibility

Problem especially apparent when using commercial simulation packages due to ease of use and lack of familiarity with underlying assumptions and restrictions

Slick graphics, animation, tables, etc. may tempt user to assign unwarranted credibility to output

Monte Carlo simulation usually requires several (perhaps many) runs at given input values

Contrast: analytical solution provides exact values

Accuracy of Result.. The Monte-Carlo simulation method is a type of sampling

procedure, thus any output is not exact but a statistical estimate whose accuracy depends on the number of missions or failures generated. For example if mission parameters are of prime important (e.g. probability of mission survival failure free) then the number of missions to be simulated is the important parameter. The number of system failures generated is not necessarily important, e.g. if in 1000 mission simulated only 5 system failures are generated, mission reliability is none the less reasonably well established. However, if MTBF(mean time between failures) estimates are the prime consideration then a sufficient number of system failures must be simulated to yield the desired accuracy.

THANK YOUThe End.