Application of Global Sensitivity Indices for Measuring the Effectiveness of Quasi-Monte Carlo Methods

The quasi-Monte Carlo (QMC) integration method can asymptotically provide a rate of convergence O(N−1), while the rate of convergence of the standard Monte Carlo (MC) method is only O(N−1/2). For a sufficiently large number of sampled points N, QMC should always outperform MC. However, for high dimensional problems such a large number of points can be infeasible. Many numerical experiments demonstrated that the advantages of QMC integration can disappear for high-dimensional problems. At the same time there are high-dimensional problems for which QMC significantly outperforms MC. Using global sensitivity indices the classification of some important classes of integrable functions is developed. It can be used for the prediction of the QMC efficiency. It is shown that the superiority of QMC over MC depends on the importance of higher-order terms in the ANOVA decomposition of an integrand. Results of numerical tests verify the prediction of the developed technique. PACS : 02.60.Jh; 02.70.Lq