MonteCarloMethod

• The Essay topic: Monte Carlo Method in Simulations

• Class:4IT496 Simulation of Systems (WS 2017/2018)

• Author: Bc. Yauheniya Andreyuk

1. Introduction

Monte Carlo method is an algorithm that rely on repeated random sampling to obtain numerical results. Essential idea of Monte Carlo is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. Monte Carlo simulation is a technique that allows to see all the possible outcomes of the decisions and assess the impact of risk, allowing better decision making under uncertainty. The technique is used by professionals in such fields as finance, project management, energy, manufacturing, engineering, research and development, insurance, oil & gas, transportation, and the environment. The simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. It shows the extreme possibilities — the outcomes of going for broke and for the most conservative decision — along with all possible consequences for middle-of-the-road decisions.

2. History

Principles of Monte Carlo Simulations the way we know it today were developed at the beginning of the 18th century. The history of Monte Carlo started with the work of a French scientist Georges Louis LeClerc, Comte de Buffon (1707-1788), who created “Buffon’s Needle.” In his work he used a method of dropping random needles on a lined background to estimate π. He proved that for a needle the same length as the distance between the lines the probability that each time a needle would intersect a line was 2/ π. He tested his theory by throwing baguettes over his shoulder on to a tile floor.

His formula was:

                                       π = 2N/X

               Where N is the number of tosses and X is the number of times a needle intersected a line.

An example of "Buffon’s Needle "can be seen here: http://www.metablake.com/pi.swf

From this simple estimation of π, the idea of using randomness in a determinative manner took off. It was revolutionary due to the fact that simulations beforehand were using a deterministic problem with statistical sampling to estimate the uncertainties in simulations.

In the 19th and 20th centuries simulation took off. “Simulation was increasingly used as an experimental means of confirming theory, analyzing data, or supplementing intuition in mathematical statistics.” (Harrison)

“Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem and statistical sampling was used to estimate uncertainties in the simulations.”(Wikipedia) Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog and solving the problem probabilistically.

These simulations were used in investigating radiation shielding and the distance that neutrons would likely travel through various materials. They were a vital simulation in the Manhattan Project even though the computational tools were underdeveloped. During their work on the Manhattan Project John von Neumann and Stanislaw Ulam named the method after the Monte Carlo Casino in Monaco.

In the 1950s they were then a crucial part of developing the Hydrogen bomb.  During this time, two large institutions financed the disseminating information on Monte Carlo simulations allowing the practical use in many other fields other than physics and chemistry. Since then Monte Carlo simulations have proven to be extremely useful in other fields such as physics, physical chemistry, operations research, business, and medical fields. Researchers now can explore complex systems or problems and find quantities that are hidden in experiments. Today, there are some disputes that simulations are over used because they remove the need for analytical or experimental approaches. Other than that Monte Carlo Simulations have been proven useful for problems that are analytically intractable and for which conducting experiments is too costly, time consuming, or impracticable.