Deterministic Versus Monte Carlo Quadrature: Theory, Experiments and the Role of Dimension
SINDANI BUKERIMANZA MOISE *
Management Information Systems, Haute Ecole de Commerce de Kinshasa, Kinshasa, DRC.
*Author to whom correspondence should be addressed.
Abstract
This paper presents a theoretical and computational comparison between classical deterministic numerical integration methods (trapezoidal, midpoint, and Simpson), Quasi–Monte Carlo (Sobol’) sampling, Smolyak sparse-grid quadrature, and the standard Monte Carlo estimator. We establish convergence rates for each method, specifying regularity assumptions and dimensional dependence, and we clarify the construction of multidimensional benchmark functions via separable tensor products. Numerical experiments are conducted on smooth, oscillatory, and weakly singular test functions in dimensions 1 ≤ d ≤ 5, with performance evaluated in terms of absolute error, total function evaluations, CPU time, and Monte Carlo variability (mean, standard error, and interquartile range). The results confirm that deterministic quadratures achieve high accuracy at low dimension but suffer from exponential cost growth with d, while Monte Carlo maintains dimensionindependent convergence at rate O(n−1/2). Quasi–Monte Carlo and sparse-grid methods provide competitive accuracy in moderate dimensions. Overall, deterministic schemes are well-suited to smooth low-dimensional integrals, whereas Monte Carlo and QMC methods remain advantageous for high-dimensional or irregular integration problems.
Keywords: Numerical integration, deterministic quadrature, Monte Carlo method, convergence analysis, computational efficiency, curse of dimensionality, error estimation, simpson’s rule, trapezoidal rule, midpoint method, high-dimensional integration, variance reduction, Quasi–Monte Carlo Sampling