Lattice rules in non-periodic subspaces of Sobolev spaces

TL;DR

This paper studies quasi-Monte Carlo integration using lattice rules in non-periodic Sobolev spaces, establishing embeddings, norm equivalences, and error bounds, and demonstrating near-optimal convergence rates with low dimensional dependence.

Contribution

It provides the first detailed analysis of embeddings and norm equivalences in non-periodic Sobolev spaces and their subspaces, solving an open problem and analyzing lattice rule errors.

Findings

Norm-equivalent subspaces identified for integer smoothness levels

Strict set inclusion established between certain Sobolev and cosine spaces for  or higher

Near-optimal convergence rates achieved with tent-transformed and symmetrized lattice rules

Abstract

We investigate quasi-Monte Carlo (QMC) integration over the $s$-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $\alpha>1$, where $\boldsymbol{\gamma}$ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter $\alpha>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $\alpha$, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $\alpha$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ as sets for $\alpha \geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case.