Optimization of Monte Carlo simulations

Abstract

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 δ = 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 δ, one may save 60%–80% of the total computing time compared to distributions with equal relative weight.