Lattice based integration algorithms: Kronecker sequences and rank-1 lattices

TL;DR

This paper establishes upper bounds on the convergence rates of lattice-based numerical integration algorithms, specifically Kronecker and rank-1 lattices, in function spaces with mixed smoothness, showing they achieve near-optimal error bounds with efficient point set generation.

Contribution

It provides new theoretical error bounds for lattice-based quasi-Monte Carlo algorithms in Besov and Sobolev spaces, highlighting the efficiency of Kronecker and rank-1 lattices.

Findings

Kronecker and rank-1 lattices achieve near-optimal error bounds.

Error bounds are expressed in terms of the Zaremba index of the lattice.

Algorithms for generating these lattices are computationally efficient.

Abstract

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $\mathring{\mathbf{B}}^s_{p,\theta}$, which also comprise the concept of Sobolev spaces of dominating mixed smoothness $\mathring{\mathbf{H}}^s_{p}$ as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from $T_N(\mathbb{Z}^d) \cap [0,1)^d$, where $T_N$ is a real invertible generator matrix of size $d$. For such rules the worst-case error can be bounded in terms of the Zaremba index of the lattice $\mathbb{X}_N=T_N(\mathbb{Z}^d)$. We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to $\log N$-factors for arbitrary smoothness $s$. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.