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 ss-variate functions. Here ss is large including s=∞s=∞. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions fkfk with only kk variables, where k=k(ε)k=k(ε) depends solely on the error demand εε and is surprisingly small when ss is sufficiently large relative to εε. This holds, in particular, for s=∞s=∞ and arbitrary εε since then k(ε)<∞k(ε)<∞ for all εε. Moreover k(ε)k(ε) does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.