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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 $s=\infty$. 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 $k=k(\varepsilon)$ depends solely on the error demand $\varepsilon$ and is surprisingly small when $s$ is sufficiently large relative to $\varepsilon$. This holds, in particular, for $s=\infty$ and arbitrary $\varepsilon$ since then $k(\varepsilon)<\infty$ for all $\varepsilon$. Moreover $k(\varepsilon)$ does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.