OPTIMIZATION OF MONTE-CARLO SIMULATIONS

Monte Carlo simulations of phase transitions and critical phenomena are hampered by long relaxation times and large fluctuations. The computational effort for the collection of data points in a simulation often varies from data point to data point by orders of magnitude. The optimal investment of computing time at different data points is a priori not clear. We propose a method to calculate distributions of statistical weight and computing time which minimize the error of the parameters to be determined. The method is discussed for the example of a finite size simulation in statistical physics. We calculate explicitly the optimal distributions for algebraic dependencies. It is shown that the optimal distributions depend on lambda = d + z - x, where d is the dimension of the system, z is the dynamical critical exponent and x is the scaling exponent of the relative variance of the data. Depending on lambda, one may save 60%-80% of the total computing time compared to distributions with equal relative weight.