## 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

Funding Information:\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section December 18, 2017; accepted for publication (in revised form) July 2, 2018; published electronically October 2, 2018. http://www.siam.org/journals/sisc/40-5/M116189.html Funding: This research was supported by the Australian Research Council (FT130100655 and DP150101770), the KU Leuven research fund (OT:3E130287 and C3:3E150478), the Taiwanese National Center for Theoretical Sciences (NCTS) --Mathematics Division, and the Statistical and Applied Mathematical Sciences Institute (SAMSI) 2017 Year-long Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics.

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