Monte Carlo Integration: A Comparison to Numerical Quadrature

Sammanfattning: Integrals are present everywhere in science, and their computation an emphasis in education. When methods of exact computation fail, a great variety of methods of approximation can step in. This project is interested in the Monte Carlo integration methods, an approach, where the integral is approximated based on the Law of Large Numbers. These methods are compared to the methods of numerical quadrature and tested on implementations, with the goal of seeing whether Monte Carlo integration could be a competitor for numerical quadrature in three and four dimensions. The comparison is made in terms of convergence, by looking at the n-th minimal error of an asymptotically optimal algorithm of each method. This shows that numerical quadrature methods have a smaller n-th minimal error for specific sets of functions in one dimension, but for sets of multivariable functions, whose smoothness are small compared to their dimension, Monte Carlo integration is a better pick.