Finite-order weights imply tractability of multivariate integration

Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can often be approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order, and ignore all groups of variables of higher order. In this paper we consider multivariate integration for the anchored and unanchored (non-periodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ε depends on s and ε. If there is no dependence on s, and only polynomial dependence on ε, we have strong tractability, whereas with polynomial dependence on both s and ε we have tractability. We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for 1 all bounded finite-order weights. In both cases, the dependence on ε is quadratic. We can improve the dependence on ε at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ε with a polynomial dependence on s whose degree is proportional to the order of the weights. We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to tractability bounds with almost linear dependence on ε. We also present explicit worst case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration.