Transformed rank-1 lattices for high-dimensional approximation

Abstract

This paper describes an extension of Fourier approximation methods for multivariate functions defined on the torus ${\mathbb{T}}^d$ to functions in a weighted Hilbert space $L_2({\mathbb{R}}^d,\omega )$ via a multivariate change of variables $\psi :{\left(-\frac{1}{2},\frac{1}{2}\right)}^d\to {\mathbb{R}}^d$. We establish sufficient conditions on $\psi$ and $\omega$ such that the composition of a function in such a weighted Hilbert space with $\psi$ yields a function in the Sobolev space $H_{\mathrm{mix}}^m({\mathbb{T}}^d)$ of functions on the torus with mixed smoothness of natural order $m\in {\mathbb{N}}_0$. In this approach we adapt algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus ${\mathbb{T}}^d$ based on single and multiple reconstructing rank-$1$ lattices. Since in applications it may be difficult to choose a related function space, we make use of dimension incremental construction methods for sparse frequency sets. Various numerical tests confirm obtained theoretical results for the transformed methods.