Very Low Truncation Dimension for High Dimensional Integration Under Modest Error Demand

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of $s$-variate functions. Here $s$ is large including $s=\mathrm{\infty }$. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions $f_k$ with only $k$ variables, where $k=k(\epsilon )$ depends solely on the error demand $\epsilon$ and is surprisingly small when $s$ is sufficiently large relative to $\epsilon$. This holds, in particular, for $s=\mathrm{\infty }$ and arbitrary $\epsilon$ since then $k(\epsilon )<\mathrm{\infty }$ for all $\epsilon$. Moreover $k(\epsilon )$ does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.