The analysis of vertex modified lattice rules in a non-periodic Sobolev space

TL;DR

This paper investigates vertex modified lattice rules within unanchored Sobolev spaces, providing error bounds and explicit formulas, especially in 2D, and demonstrating near optimal convergence rates.

Contribution

It derives a breakdown of the worst-case error for vertex modified lattice rules in unanchored Sobolev spaces, including explicit bounds and formulas in 1D and 2D cases.

Findings

Explicit error bounds for 1D and 2D cases.

Existence of lattice rules with $N^{-1} \, \log^2(N)$ error bound.

Connection between vertex modified lattice rules and classical error bounds.

Abstract

In a series of papers, in 1993, 1994 & 1996, Sloan & Niederreiter introduced a modification of lattice rules for non-periodic functions, called "vertex modified lattice rules"', and a particular breed called "optimal vertex modified lattice rules". In the 1994 paper, Niederreiter & Sloan concentrate explicitly on Fibonacci lattice rules, which are a particular good choice of 2-dimensional lattice rules. Error bounds in this series of papers were given related to the star discrepancy. In this paper we pose the problem in terms of the so-called unanchored Sobolev space, which is a reproducing kernel Hilbert space often studied nowadays in which functions have $L_2$-integrable mixed first derivatives. It is known constructively that randomly shifted lattice rules, as well as deterministic tent-transformed lattice rules and deterministic fully symmetrized lattice rules can achieve close to $O(N^{-1})$ convergence in this space, see Sloan, Kuo & Joe (2002) and Dick, Nuyens & Pillichshammer (2014) respectively. We derive a break down of the worst-case error of vertex modified lattice rules in the unanchored Sobolev space in terms of the worst-case error in a Korobov space, a multilinear space and some additional "mixture term". For the 1-dimensional case this worst-case error is obvious and gives an explicit expression for the trapezoidal rule. In the 2-dimensional case this mixture term also takes on an explicit form for which we derive upper and lower bounds. For this case we prove that there exist lattice rules with a nice worst-case error bound with the additional mixture term of the form $N^{-1} \log^2(N)$.