CS(1) Introduction to Monte Carlo Methods

History and Basic Concepts

History and Development

Monte Carlo methods were developed by Stanislaw Ulam and John von Neumann in the late 1940s, during and after World War II. These methods started as a tool for nuclear research at the Los Alamos laboratory, where they worked on extremely complex mathematical problems that were impractical to solve with traditional calculations. The name “Monte Carlo” was chosen because these methods use random and probabilistic processes, reminiscent of games of chance in Monte Carlo, Monaco.

The scientists realized they could solve some problems faster and more efficiently using random sampling and statistics than by exact deterministic calculations. With the development of ENIAC (the first electronic computer), these simulations became feasible for a wide range of scientific and mathematical problems.

Concept of Simulation with Random Numbers

Monte Carlo methods use random simulations to solve complex problems involving integrals, systems of equations, and other mathematical models. Instead of seeking an exact solution, these methods generate approximate solutions through random sampling.

The basic concept is simple:
(1) Generate random values according to a distribution (for example, values between 0 and 1 for functions on the interval [0,1]).
(2) Calculate the function of interest at these random points.
(3) Take the…