2015/01/22 by Frances Y. Kuo, Kuo, Frances Y., Dirk Nuyens +7
Computer Science · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Mathematical Approximation and Integration #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1501.05445
openalex publication_date 2015/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We further develop the \Multivariate Decomposition Method (MDM) for the\nLebesgue integration of functions of infinitely many variables\nx1,x2,x3,\… with respect to a corresponding product of a one\ndimensional probability measure.\n Although a number of concepts of infinite-dimensional integrals have been\nused in the literature, questions of uniqueness and compatibility have mostly\nnot been studied. We show that, under appropriate convergence conditions, the\nLebesgue integral equals the `anchored' integral, independently of the anchor.\n The MDM assumes that point values of f_ mathfraku are available for\nimportant subsets mathfraku, at some known cost. In this paper we\nintroduce a new setting, in which it is assumed that each f_ mathfraku\nbelongs to a normed space F_ mathfraku, and that bounds\nB_ mathfraku on \‖f_ mathfraku\‖_F_ mathfraku are known. This\ncontrasts with the assumption in many papers that weights\n\γ_ mathfraku, appearing in the norm of the infinite-dimensional\nfunction space, are somehow known. Often such weights \γ_ mathfraku\nwere determined by minimizing an error bound depending on the\nB_ mathfraku, the \γ_ mathfraku \and the chosen\nalgorithm, resulting in weights that depend on the algorithm. In contrast, in\nthis paper only the bounds B_ mathfraku are assumed known.\n We give two examples in which we specialize the MDM: in the first case\nF_ mathfraku is the | mathfraku|-fold tensor product of an anchored\nreproducing kernel Hilbert space, and in the second case it is a particular\nnon-Hilbert space for integration over an unbounded domain.\n