Chaotic Monte Carlo computation: a dynamical effect of random-number generations

It is shown that superefficient Monte Carlo computations can be carried out by using chaotic dynamical systems as non-uniform random-number generators. Here superefficiency means that the expectation value of the square of the error decreases to 0 as 1/N^{2} with N successive observations for N-> infinity, whereas the conventional Monte Carlo simulation gives the square of the error in the order 1/N. The order of N in the error convergence speed of superefficient Monte Carlo computations does not depend on the dimensionality of the problems. 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 of chaotic dynamical variables.