Quasi-random integration in high dimensions

In this paper, we show that the Sobol’ and Richtmyer sequences can be effectively used for numerical integration of functions having up to 1000 variables. The results of integration obtained with the two sequences are compared and the parameters C   and αα from the convergence model C/NαC/Nα are estimated, where N is the number of points used. For all the tests done, the Sobol’ sequence demonstrated somewhat better convergence, but for many practical values of N the relative error is higher than for Richtmyer sequences due to the large value of C. Constructing Sobol’ sequences also takes considerably more time than constructing Richtmyer sequences.