  We study numerical integration of functions depending on an infinite number
of variables. We provide lower error bounds for general deterministic linear
algorithms and provide matching upper error bounds with the help of suitable
multilevel algorithms and changing dimension algorithms.
  More precisely, the spaces of integrands we consider are weighted reproducing
kernel Hilbert spaces with norms induced by an underlying anchored function
space decomposition. Here the weights model the relative importance of
different groups of variables. The error criterion used is the deterministic
worst case error. We study two cost models for function evaluation which depend
on the number of active variables of the chosen sample points, and two classes
of weights, namely product and order-dependent (POD) weights and the newly
introduced weights with finite active dimension. We show for these classes of
weights that multilevel algorithms achieve the optimal rate of convergence in
the first cost model while changing dimension algorithms achieve the optimal
convergence rate in the second model.
  As an illustrative example, we discuss the anchored Sobolev space with
smoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlo
multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based
on higher-order polynomial lattice rules.
