On the convergence rate of the component-by-component construction of good lattice rules

Abstract

We prove error bounds on the worst-case error for integration in certain Korobov and Sobolev spaces using rank-1 lattice rules with generating vectors constructed by the component-by-component algorithm. For a prime number of points n a rate of convergence of the worst-case error for multivariate integration in Korobov spaces of O(n −α/2+δ) , where α>1 is a parameter of the Korobov space and δ is an arbitrary positive real number, has been shown by Kuo. First we improve the constant of this error bound. Further, we prove an error bound which shows that the rate of convergence is optimal up to a power of log n for prime n. These error bounds are then generalised to the case where the number of points is not a prime number. Numerical results comparing the worst-case errors and the error bounds are presented.