On strong tractability of weighted multivariate integration

Abstract:

We prove that for every dimension $s$ and every number $n$ of points, there exists a point-set $\mathcal {P}_{n,s}$ whose $\boldsymbol \gamma$-weighted unanchored $L_{\infty }$ discrepancy is bounded from above by $C(b)/n^{1/2-b}$ independently of $s$ provided that the sequence $\boldsymbol \gamma =\{\gamma _k\}$ has $\sum _{k=1}^\infty \gamma _k^a<\infty$ for some (even arbitrarily large) $a$. Here $b$ is a positive number that could be chosen arbitrarily close to zero and $C(b)$ depends on $b$ but not on $s$ or $n$. This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals $\int _Df(\mathbf {x}) \rho (\mathbf {x}) d\mathbf {x}$ over unbounded domains such as $D=\mathbb {R}^s$. It also supplements the results that provide an upper bound of the form $C\sqrt {s/n}$ when $\gamma _k\equiv 1$.