Infinite-dimensional integration and the multivariate decomposition method

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 i 1 , … , x i k . 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 ‖ F u 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.