Using Deterministic Chaos for Superefficient Monte Carlo Simulations

Monte Carlo (MC) simulation methods are widely used to solve complex engineering and scientific problems. Unlike other deterministic methods, MC methods use statistical sampling to produce approximate solutions. As the processed sample size N growths, the uncertainty of the solution is reduced. It is well known that the mean-square approximation error decreases as 1/N. However, for large problems like high-dimensional integrations and computationally intensive simulations, MC methods may take months or even years to obtain a solution with acceptable tolerance. The Super-Efficient (SE) Monte Carlo simulation method, originated by Umeno, produces a solution whose approximation error decreases as fast as 1/N². However, it only applies to a small class of problems possessing certain properties. We describe an approximate SE Monte Carlo simulation method that is applicable to a wider class of problems than the original SE method, and yields a convergence rate as fast as 1/N^α for 1 ≤ α ≤ 2.