Infinite-dimensional integration and the multivariate decomposition method

Abstract

We further develop the \emph{Multivariate Decomposition Method} (MDM) for the Lebesgue integration of functions of infinitely many variables $x_1,x_2,x_3,\ldots$ with respect to a corresponding product of a one dimensional probability measure. 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. The MDM assumes that point values of $f_{\mathfrak{u}}$ are available for important subsets ${\mathfrak{u}}$, at some known cost. In this paper we introduce a new setting, in which it is assumed that each $f_{\mathfrak{u}}$ belongs to a normed space $F_{\mathfrak{u}}$, and that bounds $B_{\mathfrak{u}}$ on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ are known. This contrasts with the assumption in many papers that weights $γ_{\mathfrak{u}}$, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights $γ_{\mathfrak{u}}$ were determined by minimizing an error bound depending on the $B_{\mathfrak{u}}$, the $γ_{\mathfrak{u}}$ \emph{and} the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds $B_{\mathfrak{u}}$ are assumed known. We give two examples in which we specialize the MDM: in the first case $F_{\mathfrak{u}}$ is the $|{\mathfrak{u}}|$-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain.