# Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha-1/2})$ error independently of the dimension

## TL;DR

This paper introduces a random lattice rule algorithm for high-dimensional integration that achieves near-optimal error rates independent of dimension, using a simple and implementable approach based on Bakhvalov's randomization technique.

## Contribution

It demonstrates that random rank-1 lattice rules can attain nearly optimal convergence rates for weighted Sobolev spaces, with error bounds independent of dimension, and simplifies implementation compared to existing methods.

## Key findings

- Achieves error rate of (n^{-\u03b1-1/2}) for <
- Error bounds are independent of the problem dimension under certain conditions
- Method is easier to implement than Frolov's method in high dimensions

## Abstract

We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where the functions are continuous and periodic when $\alpha>1/2$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. For the case $\alpha>1/2$, we prove that the algorithm achieves almost the optimal order of convergence of $\mathcal{O}(n^{-\alpha-1/2})$, where the implied constant is independent of the dimension~$d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha}<\infty$. The same rate of convergence holds for the more general case $\alpha>0$ by adding a random shift to the lattice rule with random $n$. This shows, in particular, that the exponent of strong tractability in the randomized setting equals $1/(\alpha+1/2)$, if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-$1$ lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.