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

TL;DR

This paper introduces a simple, randomised lattice rule algorithm for high-dimensional numerical integration in Korobov spaces, achieving near optimal convergence rates with a fixed generating vector and random choice of the number of evaluations.

Contribution

The paper presents a novel randomised lattice rule method with fixed generating vectors that attains near optimal convergence rates for high-dimensional integrals.

Findings

Error bounds of order $O(n^{-eta})$ for the algorithm's expected error.

Dimension-independent constants under certain weight conditions.

Numerical experiments confirm theoretical convergence rates.

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 $O(n^{-\alpha - 1/2 + \delta})$, 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 \emph{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.