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In this paper we focus on efficient implementations of the MultivariateDecomposition Method (MDM) for approximating integrals of $\infty$-variatefunctions. Such $\infty$-variate integrals occur for example as expectations inuncertainty quantification. Starting with the anchored decomposition $f =\sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{u}$, where the sum is over allfinite subsets of $\mathbb{N}$ and each $f_\mathfrak{u}$ depends only on thevariables $x_j$ with $j\in\mathfrak{u}$, our MDM algorithm approximates theintegral of $f$ by first truncating the sum to some `active set' and thenapproximating the integral of the remaining functions $f_\mathfrak{u}$term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures.The anchored decomposition allows us to compute $f_\mathfrak{u}$ explicitly byfunction evaluations of $f$. Given the specification of the active set andtheoretically derived parameters of the quadrature rules, we exploit structuresin both the formula for computing $f_\mathfrak{u}$ and the quadrature rules todevelop computationally efficient strategies to implement the MDM in variousscenarios. In particular, we avoid repeated function evaluations at the samepoint. We provide numerical results for a test function to demonstrate theeffectiveness of the algorithm.

Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals