Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration

Abstract

We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the d-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget n of a maximum allowed number of function evaluations, we uniformly pick a prime p in the range $n/2<p\le n$. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form $O\left(n^{-\alpha -1/2+\delta }\right)$, with $\delta >0$, for a Korobov space with smoothness $\alpha >1/2$ and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors offline ahead of time at cost $O(d{n}^4/\ln n)$ when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.