Introduction to Monte Carlo methods

Solving many scientific problems involves calculating sums or integrals, or solving equations or optimization problems. Direct calculation techniques, also known as analytical techniques, are very quickly outstripped by the complexity of the models: they often require assumptions that are too strong, so that they cannot be applied, or, as in the case of calculating sums, the number of operations required may be too great to be carried out in a reasonable time. In such cases, approximation methods must be used. However, these also require strong assumptions, albeit not as strong as for analytical methods. What's more, these methods quickly prove ineffective as the mathematical dimension of the problem increases. To illustrate this phenomenon, table shows the convergence speeds of the usual quadrature rules for calculating multiple integrals (n represents the number of...