Constructing Good Lattice Rules with Millions of Points

We develop an algorithm for the construction of randomly shifted rank-1 lattice rules in d-dimensional weighted Sobolev spaces with a significantly reduced construction cost. The results shown here are an extension of earlier results by the present authors. In this new algorithm, the number of quadrature points n is a product of r distinct prime numbers p 1,…,p r. This allows us to reduce the construction cost to O(n(p 1 + … +p r)d 2), which represents a significant reduction, especially for large n. The constructed rules achieve a worst-case error bound with a rate of convergence of O(n(p 1 + δ p 2 -1/2 ... p r -1/2 ) for any δ > 0. Numerical experiments were carried out for r = 2, 3, 4 and 5. The results demonstrate that it can be advantageous to choose n as a product of up to 5 primes.