Lattice rules in non-periodic subspaces of Sobolev spaces

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 $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $α>1$, where $\boldsymbolγ$ 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 $α>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $α$, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $α$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ as sets for $α\geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbolγ,s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. 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.