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Monte Carlo Techniques


Introduction

Monte Carlo methods are numerical methods based on random sampling. Scientists and engineers use simulation methods to solve problems which are insoluble by analytical techniques. Monte Carlo methods which make use of probabilistic simulations are frequently used in areas such as numerical integration, complex scheduling, queueing networks, and large-dimensional simulations. It provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer.

History

The method is called after the city in the Monaco principality, because of a roulette, a simple random number generator. The name and the systematic development of Monte Carlo methods dates from about 1944.
In the second half of the nineteenth century a number of people performed experiments, in which they threw a needle in a haphazard manner onto a board ruled with parallel straight lines and inferred the value of Pi = 3.14 from observations of the number of intersections between needle and lines. The test was called the Buffon's Needle Test. In early part of the twentieth century, British statistical schools indulged in a fair amount of unsophisticated Monte Carlo work. Most of this work was rarely used for research or discovery. The real use of Monte Carlo methods as a research tool stems from work on the atomic bomb during the second world war. This work involved a direct simulation of the probabilistic problems concerned with random neutron diffusion in fissile material.

Estimating the value of "Pi" by Monte Carlo Methods

PiEstimation(137kB) The adjoining figure shows a circle of radius "r" inside a square of side "2r". Let us throw darts into the square at random. It is apparent that of the total number of darts that are thrown, the number of darts that hit the circle (the unshaded part) is proportional to the area of the circle. The ratio of the area of the circle to the area of the square is also Pi/4, as explained in the diagram.
If we actually do the experiment, it takes a sufficiently large number of throws for the ratio 4 * M/N to get a decent value of Pi. The applet below gives a simple demonstration of the method.

Please wait for the image/applet to load. It may take some time depending on the network traffic.

The number of throws has been limited (1000 - 20,000) to see fast results. The more the number of throws, the more accurate the value of Pi.
Please enter the number of iterations in the text box and hit the "Start" button.
Happy throwing !!!


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