On the optimization of Monte-Carlo simulations

The optimal planning of Monte-Carlo simulations is studied. It is assumed that (i) the aim of the simulations is to calculate the value of a certain parameter of a model function as accurately as possible; (ii) the simulations are performed at different values of the control parameter L; (iii) the parameters of the model function are calculated by the means of least-square fit. It is shown that the standard deviation of the outcome achieves minimum when the number of test points (i.e. different values of the parameter L used in simulations) equals the number n of unknown parameters in the model function. For simpler model functions (n less than or equal to 4), the test points and the respective weights describing the distribution of the computer time can be found analytically. As an example, the calculation of fractal dimensions is discussed.