Monte Carlo methods

When we speak of Monte Carlo, we refer to a large group of numerical methods for solving physics problems by means of random sampling. In particular, the Monte Carlo method allows us to simulate any process whose development is affected by random factors, and many mathematical problems not affected by any random influences can be connected to the Monte Carlo method by an artificially constructed probabilistic model. Monte Carlo methods thus form the topic of this chapter. Since Monte Carlo methods are stochastic processes, we first review the required probability and statistics topics. We then develop the basic techniques. These techniques are used to perform integration and sample distributions using the basic methods of uniform, importance, and stratified sampling. Finally, the method of Markov-chain Monte Carlo is introduced.