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

Alexander D. Gilbert, Frances Y. Kuo, Dirk Nuyens, Grzegorz W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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 = ΣU⊂N fu, where the sum is over all finite subsets of ℕ and each fu depends only on the variables xj 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 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.

Original languageEnglish
Pages (from-to)A3240-A3266
JournalSIAM Journal on Scientific Computing
Volume40
Issue number5
DOIs
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

Keywords

  • Infinite-variate integral
  • Multivariate decomposition method
  • Quadrature
  • Quasi-Monte Carlo
  • Smolyak's method
  • Sparse grids

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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