next up previous contents
Next: Importance Sampling Up: Quantum Monte Carlo Methods Previous: The Variational Principle

Monte Carlo Methods

Monte Carlo methods are a way of using random numbers to perform numerical integrations. By way of example consider the integral


There are many quadrature methods, with varying degrees of accuracy, which can be used to evaluate this integral. The trapezium rule and Simpson's method (see ``Numerical Recipes'', [18]) are both quadrature methods which involve evaluating f(x) at evenly spaced points, tex2html_wrap_inline6081 , on a grid. A weighted average of these values tex2html_wrap_inline6083 gives an estimate of the integral


where the tex2html_wrap_inline6085 are the weights. The weights and the sampling points are different for different methods of quadrature but all the methods sample the function f(x) using pre-determined weights and sampling points.

Monte Carlo methods do not use specific sampling points but instead we choose points at random. The Monte Carlo estimate of the integral is then,


where the tex2html_wrap_inline6081 are randomly sampled points and tex2html_wrap_inline6091 is the arithmetic mean of the values of the function f(x) at the sampling points. The standard deviation of the mean is given by




gives an estimate of the statistical error in the Monte Carlo estimate of the integral. Note that the error goes as tex2html_wrap_inline6095 , independent of the dimensionality of the integral.

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996