Very low truncation dimension for high dimensional integration under modest error demand: Dedicated to the memory of Joseph F. Traub (1932–2015)

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of s-variate functions. Here s is large including 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 f_k with only k variables, where k=k(ε) depends solely on the error demand ε and is surprisingly small when s is sufficiently large relative to ε. This holds, in particular, for s=∞ and arbitrary ε since then k(ε)<∞ for all ε. Moreover 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.