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 = ∑ᵤ fᵤ, where the sum is over all finite subsets of N and each fᵤ depends only on the variables xⱼ with j, 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 fᵤ term-by-term using Smolyak or (randomized) quasi--Monte Carlo quadratures. The anchored decomposition allows us to compute fᵤ 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 fᵤ 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.