The Analysis of Vertex Modified Lattice Rules in a Non-periodic Sobolev Space

In a series of papers, in 1993, 1994 and 1996, Ian Sloan together with Harald 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”, see Numerical Integration IV (Birkhäuser 1993) pp. 253–265, J Comput Appl Math 51(1):57–70, 1994, and Comput Math Model 23(8–9):69–77, 1996. These are like standard lattice rules but they distribute the point at the origin to all corners of the unit cube, either by equally distributing the weight and so obtaining a multi-variate variant of the trapezoidal rule, or by choosing weights such that multilinear functions are integrated exactly. In the 1994 paper, Niederreiter and 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 et al. (Math Comput 71(240):1609–1640, 2002) and Dick et al. (Numer Math 126(2):259–291, 2014) respectively, where possible log s ( N ) $$\log ^{s}(N)$$ terms are taken care of by weighted function spaces. 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 ) $$N^{-1}\log ^{2}(N)$$ .