Parallel Monte Carlo Method

Introduction

Monte Carlo method is very general. use random numbers to approximate solutions to

problems. especially useful for simulating systems with

many coupled degrees of freedom simulate procedures with a large number of

inputs. We can apply the method in everything from

economics to nuclear physics to regulating the flow of traffic.

Common uses

Traffic flow

Financial analysis

Computer Graphics

Nuclear reactor design

Molecular dynamics

Radiation cancer therapy

General Guidelines

Define a domain of possible inputs.

Generate inputs randomly from a probability

distribution over the domain.

Perform a deterministic computation on the

inputs.

Aggregate the results.

Example

Lets approximate

Given that function and 1x1 square have a ratio of

areas that is 1/2

Example 1/4for i=1:N; x=rand; y=rand; F(i,1)=x; F(i,2)=y; if(x<=y) count=count+1; endendsol = count/N;

Example 2/4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Number of samples is:100 The Solution is:0.520000

x

y

Example 3/4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Number of samples is:1000 The Solution is:0.486000

x

y

Example 4/4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Number of samples is:10000 The Solution is:0.497900

x

y

Method summary

So Generally, the more iterations of the Monte Carlo

simulation, the better the approximation will be.

The problem: It requires an intensive computing

Can we parallelize ?

Method flow chart - serial

Method flow chart - parallel

SpeedUp

Conclusions

Monte Carlo Algorithms are very easy to convert to parallel algorithms.

When scaling to larger numbers of parallel CPU’s we see a smooth decrease in the time spent per timestep for a given simulation.

Questions?