Error analysis of an adaptive Monte Carlo method for numerical integration

A new adaptive technique for Monte Carlo (MC) integration is proposed and studied. An error analysis is given. It is shown that the error of the numerical integration depends on the smoothness of the integrand. A superconvergent adaptive method is presented. The method combines the idea of separation of the domain into uniformly small subdomains with the Kahn approach of importance sampling. An estimation of the probable error for functions with bounded derivatives is proved. This estimation improves the existing results. A simple adaptive Monte Carlo method is also considered. It is shown that for large dimensions d the convergence of the superconvergent adaptive MC method goes asymptotically to O(n1/2), which corresponds to the convergence of the simple adaptive method. Both adaptive methods – superconvergent and simple – are used for calculating multidimensional integrals. Numerical tests are performed on the supercomputer CRAY Y-MP C92A. It is shown that for low dimensions (up to d=5) the superconvergent adaptive method gives better results than the simple adaptive method. When the dimension increases, the simple adaptive method becomes better. One needs several seconds for evaluating 30-d integrals using the simple adaptive method, while the evaluation of the same integral using Gaussian quadrature will need more than 106 billion years if CRAY Y-MP C92A is used.