## 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 = 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.

Original language | English |
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Pages (from-to) | A3240-A3266 |

Journal | SIAM Journal on Scientific Computing |

Volume | 40 |

Issue number | 5 |

DOIs | |

State | Published - 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