Efficient implementations of the multivariate decomposition method for approximating infinite-variate integrals

In this paper we focus on efficient implementations of the multivariate decomposition method (MDM) for approximating integrals of ∞-variate functions. Such ∞-variate integrals occur, for example, as expectations in uncertainty quantification. Starting with the anchored decomposition f = Σ_U⊂N f_u, where the sum is over all finite subsets of ℕ and each f_u depends only on the variables x_j with j ∈ 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_u term-by-term using Smolyak or (randomized) quasi-Monte Carlo quadratures. The anchored decomposition allows us to compute f_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_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.