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

Abstract

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(n3d2) operations. This may be reduced to O(n3d) operations at the expense of O(n2) 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.