A Comparative Analysis of Monte Carlo Integrations and Numerical Quadrature for Multi-Dimensional Functions

Abstract

Multidimensional integrals can be used in physics, machine learning, and engineering. However, they are difficult to solve correctly because the integrands are complex and the cost of calculation increases with the dimensionality. This research compares Monte Carlo Integration (MCI), Quasi- Monte Carlo (QMC) and Numerical Quadrature (NQ) techniques of solving multi-dimensional integrals that are chosen as benchmark functions. The objective of this research is to compare the methods with the benchmark function as the accuracy, computational efficiency, and convergence rate measure. Both MCI and QMC are consistently accurate, and MCI is more successful in high-dimensional problems because of its dimension-independent convergence. QMC is better at low dimensions but not higher-dimensional problems. NQ performs almost perfectly in low dimensionality and is computationally efficient with increasing dimensionality. Four more benchmark functions are used to evaluate the convergence behaviour, including smooth exponential, poly-separable, oscillatory, and nonsmoothed indicator functions. The findings shows that MCI is converging with a steady rate for all integrands. QMC is seen to be more effective in smoothing and separable functions, whilst convergence is compromised in oscillatory and nonsmoothed integrands. NQ shows second-order convergence to smooth integrals in low-dimensions, but it cannot do nonsmoothed functions. These results give an idea of the advantages and disadvantages of each technique based on the dimension and nature of integrand.