Monte Carlo methods have been used extensively in recent years to quantify
uncertainty in real-life applications, ranging from nuclear reactors and
aviation through to finance. We introduce a new Monte Carlo based sampling
technique, known as Multi-Index Monte Carlo (MIMC), that allows us to gain
further variance reduction in comparison with the Multilevel Monte Carlo
method (MLMC). Ultimately, this is achieved by using a multi-dimensional set
of hierarchies and by then taking higher order differences. We begin by
introducing special cases of MIMC, namely the naive Monte Carlo and MLMC
methods. From there we will present the underlying ideas behind MIMC, before
showing how different choices of the set of hierarchies can reduce
computational cost whilst maintaining accuracy.

Based on a paper by Abdul-Lateef Haji-Ali et al. (2015).