Estimation of multidimensional integrals: is Monte Carlo the best method?

The effectiveness of a deterministic algorithm recently proposed by Wozniakowski (1991) for the numerical estimation of high-dimensional integrals based on so-called optimal sampling points is studied by using it to compute virial coefficients for the hard-sphere-fluid. Although the algorithm is inherently more efficient than the familiar Monte Carlo method for sufficiently large samples, in practice this advantage diminishes rapidly with increasing dimensionality. For the viral coefficient problem, the authors find the new algorithm to be noticeably more efficient than crude Monte Carlo methods for dimensions less than 10 and of comparable efficiency for larger dimensionality up to 14.