蒙特卡罗方法并行计算

Monte Carlo Methods in Parallel Computing

Chuanyi Ding ding@

Eric Haskin haskin@

Copyright by UNM/ARC

November 1995

Outline

What Is Monte Carlo?

Example 1 - Monte Carlo Integration To Estimate Pi

Example 2 - Monte Carlo solutions of Poisson's Equation

Example 3 - Monte Carlo Estimates of Thermodynamic

Properties

General Remarks on Parallel Monte Carlo

What is Monte Carlo?

• A powerful method that can be applied to otherwise intractable problems

• A game of chance devised so that the outcome from a large number of plays is the value of the quantity sought

•On computers random number generators let us play the game

•The game of chance can be a direct analog of the process being studied or artificial

•Different games can often be devised to solve the same problem •The art of Monte Carlo is in devising a suitably efficient game.

Different Monte Carlo Applications

Radiation transport

Operations research

Nuclear criticality

Design of nuclear reactors

Design of nuclear weapons

Statistical physics

Phase transitions

Wetting and growth of thin films Atomic wave functions and eigenvalues Intranuclear cascade reactions Thermodynamic properties

Long chain coiling polymers

Reaction kinetics

Partial differential equations

Large sets of linear equations Numerical integration

Uncertainty analysis

Development of statistical tests

Cell population studies

Combinatorial problems

Search and optimization

Signal detection

WarGames

Probability Theory

•Probability Density Function

•Expection Value (Mean)

•Variance and Standard Deviation

Sampling a Cumulative Distribution Function

Fundamental Theorem

•Sample Mean, a Monte Carlo estimator of the expection value

•Fundamental theorem of Monte Carlo

Statistical Error in MC Estimator

•Variance and Standard Deviation of Estimator

•Central Limit Theorem

Example 1 - Monte Carlo Integration to Estimate Pi • A simple example to illustrate Monte Carlo principals

•Accelerate convergence using variance reduction techniques o Use if expectation values

o Importance sampling

•Adapt to a parallel environment

Monte Carlo Estimate of Pi

When to Consider Monte Carlo Integration

•Time required for Monte Carlo Integration to a standard error epsilon

•Time required for numerical integration in n dimensions with trunction error of order h to the power k. (Note: k for Simpson's 1/3 rule is 4)

•Rule of Thumb

Serial Monte Carlo Algoritm for Pi

Read N

Set SumG = 0.0

Do While i < N

Pick two random numbers xi and yi

If (xi*xi + yi*yi £ 1) then

SumG = SumG + 1

Endif

Enddo

Gbar = SumG / N

SigGbar = Sqrt(Gbar - Gbar2)

Print N, Gbar, SigGbar