  In this paper we focus on efficient implementations of the Multivariate
Decomposition Method (MDM) for approximating integrals of $\infty$-variate
functions. Such $\infty$-variate integrals occur for example as expectations in
uncertainty quantification. Starting with the anchored decomposition $f =
\sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{u}$, where the sum is over all
finite subsets of $\mathbb{N}$ and each $f_\mathfrak{u}$ depends only on the
variables $x_j$ with $j\in\mathfrak{u}$, our MDM algorithm approximates the
integral of $f$ by first truncating the sum to some `active set' and then
approximating 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 by
function evaluations of $f$. Given the specification of the active set and
theoretically derived parameters of the quadrature rules, we exploit structures
in both the formula for computing $f_\mathfrak{u}$ and the quadrature rules to
develop computationally efficient strategies to implement the MDM in various
scenarios. In particular, we avoid repeated function evaluations at the same
point. We provide numerical results for a test function to demonstrate the
effectiveness of the algorithm.
