Chaotic Monte Carlo Computation: A Dynamical Effect of Random-Number Generations

Chaotic maps with absolutely continuous invariant probability measures are implemented as random-number generators for Monte Carlo computation. We observe that such Monte Carlo computation based on chaotic random-number generators yields sometimes unexpected dynamical dependency behavior which cannot be explained by usual statistical arguments. Furthermore, we find that superefficient Monte Carlo computation with O (1/ N 2 ) mean square error can be carried out as an extreme case of such dynamical dependency behavior. Here, such superefficiency sharply contrasts with the conventional Monte Carlo simulation with O (1/ N ) mean square error. By deriving a necessary and sufficient condition for the superefficiency, it is shown that such high-performance Monte Carlo simulations can be carried out only if there exists a strong correlation with chaotic dynamical variables. Numerical calculation illustrates this dynamics dependency and the superefficiency of various chaotic Monte Carlo computations.