## Abstract

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x_{1},x_{2},x_{3},… with respect to a corresponding product of a one dimensional probability measure. The method is designed for functions that admit a dominantly convergent decomposition f=∑_{u}f_{u}, where u runs over all finite subsets of positive integers, and for each u={i_{1},…,i_{k}} the function f_{u} depends only on x_{i1},…,x_{ik}. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the ‘anchored’ integral, independently of the anchor. For approximating the integral, the MDM assumes that point values of f_{u} are available for important subsets u, at some known cost. In this paper, we introduce a new setting, in which it is assumed that each f_{u} belongs to a normed space F_{u}, and that bounds B_{u} on ‖f_{u}‖_{Fu} are known. This contrasts with the assumption in many papers that weights γ_{u}, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γ_{u} were determined by minimizing an error bound depending on the B_{u}, the γ_{u} and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper, only the bounds B_{u} are assumed to be known. We give two examples in which we specialize the MDM: in the first case, F_{u} is the |u|-fold tensor product of an anchored reproducing kernel Hilbert space; in the second case, it is a particular non-Hilbert space for integration over an unbounded domain.

Original language | English |
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Pages (from-to) | 217-234 |

Number of pages | 18 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 326 |

DOIs | |

State | Published - Dec 15 2017 |

### Bibliographical note

Funding Information:The research of the first and fourth authors was supported by the Australian Research Council under projects DP110100442, FT130100655, and DP150101770. The second author was partially supported by the Research Foundation Flanders (FWO) and the KU Leuven research fund OT:3E130287 and C3:3E150478. The research of the third author was supported by the National Science Centre, Poland, based on the decision DEC-2013/09/B/ST1/04275.

Publisher Copyright:

© 2017 Elsevier B.V.

## Keywords

- Cubature
- Infinite-dimensional
- Quadrature

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics