On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

We develop and justify an algorithm for the construction of quasi-Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found ``component-by-component'': the ($d+1$)-th component of the generator vector and the shift are obtained by successive $1$-dimensional searches, with the previous $d$ components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for $n$-point rules with $n$ prime and all dimensions 1 to $d$ requires a total cost of $O(n^3d^2)$ operations. This may be reduced to $O(n^3d)$ operations at the expense of $O(n^2)$ storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and $n$ up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.