While science and mathematics are two distinct areas of study, their applications often collide. In this era of computers and technology, the use of mathematical analysis and simulation to solve scientific problems is becoming increasingly more prevalent. While some scientific problems can be solved analytically using simple equations, others require significantly more complex operations (Wittwer, 2004a; Najarian, 2018a).
In general, scientific problems can be categorized into three distinct mathematical groups, each of which is solved in a different way (Najarian, 2018a). The first group of problems can be solved analytically (Najarian, 2018a); the scientific question can be described in the form of a series of equations that can be solved exactly, using basic mathematical techniques. These simple models are usually deterministic, meaning that given a certain set of inputs the same outputs are obtained each time the problem is solved (Wittwer, 2004a) (see Figure 1). In the second case a problem can be described in the form of equations, but cannot be solved analytically (Najarian, 2018a). Approximations can be used to find solutions to these problems, though no exact conclusions can be calculated. The final mathematical group occurs when a scientific question can be neither analytically described, nor solved, either because it is too complex or because it is of a probabilistic nature and may involve probability distributions as inputs, rather than simple numbers (Najarian, 2018a). The latter group, which will be further explored, requires more complex analysis, such as the Monte Carlo Method (Najarian, 2018a).
The Monte Carlo Method, named by S. Ulam and N. Metropolis after the city in Monaco (Wittwer, 2004a), is a mathematical simulation technique that generates results based on random numbers (Gentle, 2010) to solve problems of probabilistic nature (Lafortune, 1995). It takes scientific problems which are known to be dependent on probability distributions and breaks them down into deterministic models (Wittwer, 2004a). It is then capable of generating pseudorandom numbers (Gentle, 2010) based on the probability distribution on which the problem is dependent, to use as inputs for the model (Wittwer, 2004a). An important aspect of Monte Carlo is its ability to repeat randomized events extensively to generate a distribution showing all possible outcomes (Kroese et al., 2014). This technique allows the Monte Carlo Method to be used in the analysis of virtually any scientific question, including those that cannot be represented by or solved with analytical formulas (Najarian, 2018a).
Perhaps the best way to illustrate this method of mathematical analysis is to use a very simple example. Consider the behaviour of a coin as it tossed numerous times in a row. The Monte Carlo Method is able to model this coin toss and compute any statistics that result from the theoretical experiment (Najarian, 2018a) (see Figure 2). In doing this, it essentially duplicates what would occur in real life so that the experiment does not have to be conducted by hand, but the results still represent what would actually occur (Najarian, 2018a).
The Monte Carlo Method is becoming increasingly more important in our modern society, as we continue to ask more mathematically complex scientific questions while diving further into the worlds of coding and mathematical modelling (Najarian, 2018a). While controversial in the past, due to its seemingly infinite applications (Lafortune, 1995), it’s relevance to all spectrums of science has proven essential to scientific discoveries ranging from macroscopic phenomena, such as rocket launchings, to microscopic simulations, such as Brownian motion (Najarian, 2018a). Though the Monte Carlo Method has already aided in numerous important discoveries, its widespread use leads scientists to wonder where its application will arise next.
Works Cited
Gentle, J.E., 2010. Computational Statistics. International Encyclopedia of Education, 3, pp.93–97. https://doi.org/10.1016/B978-0-08-044894-7.01316-6.
Kroese, D.P., Brereton, T., Taimre, T. and Botev, Z.I., 2014. Why the Monte Carlo method is so important today. Wiley Interdisciplinary Reviews: Computational Statistics, 6(6), pp.386–392. https://doi.org/10.1002/wics.1314
Lafortune, E.P., 1995. Mathematical Models and Monte Carlo Algorithms for Physically Based Rendering. [thesis] Mathematical Models and Monte Carlo Algorithms for Physically Based Rendering. Flemish Institute for the Promotion of Scientific and Technological Research in the Industry. Available at: <https://pdfs.semanticscholar.org/36fc/659c321e4f67ed274117334ddab41353bb91.pdf> [Accessed 21 Feb. 2019].
Najarian, J.P., 2018a. Monte Carlo Techniques. In: Salem Press Encyclopedia of Science. [online] Salem Press, pp.1–6. Available at: McMaster University library website <https://library.mcmaster.ca/> [Accessed 21 Feb. 2019].
Najarian, J.P., 2018b. Code applying Monte Carlo Method to the behaviour of 1000 coin tosses. Salem Press Encyclopedia of Science. Available at: McMaster University library website <https://library.mcmaster.ca/>[Accessed 21 Feb. 2019].
Wittwer, J., 2004a. Monte Carlo Simulation Basics. [online] Vertex42. Available at: <https://www.vertex42.com/ExcelArticles/mc/MonteCarloSimulation.html> [Accessed 21 Feb. 2019].
Wittwer, J., 2004b. A parametric deterministic model maps a set of input variables to a set of output variables. Vertex42. Available at: <https://www.vertex42.com/ExcelArticles/mc/MonteCarloSimulation.html> [Accessed 21 Feb. 2019].