A comparison between (quasi-)Monte Carlo and cubature rule based methods for solving high-dimensional integration problems

Algorithms for estimating the integral over hyper-rectangular regions are discussed. Solving this problem in high dimensions is usually considered a domain of Monte Carlo and quasi-Monte Carlo methods, because their power degrades little with increasing dimension. These algorithms are compared to integration routines based on interpolatory cubature rules, which are usually only used in low dimensions. Adaptive as well as non-adaptive algorithms based on a variety of rules result in a wide range of different integration routines. Empirical tests performed with Genz’s test function package show that cubature rule based algorithms can provide more accurate results than quasi-Monte Carlo routines for dimensions up to s = 100.