Volume 53, pp. 239-282, 2020.

Transformed rank-1 lattices for high-dimensional approximation

Robert Nasdala and Daniel Potts

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 $\psi :{(-\frac{1}{2},\frac{1}{2})}^d\to {\mathbb{R}}^d$. We establish sufficient conditions for $\psi$ and $\omega$ such that the composition of a function in such a weighted Hilbert space with $\psi$ yields a function in the Sobolev space $H_{\mathrm{m}\mathrm{i}\mathrm{x}}^m({\mathbb{T}}^d)$ 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 the obtained theoretical results for the transformed methods.

Full Text (PDF) [1.3 MB], BibTeX , DOI: 10.1553/etna_vol53s239

Key words

approximation on unbounded domains, change of variables, sparse multivariate trigonometric polynomials, lattice rule, multiple rank-$1$ lattice, fast Fourier transform

AMS subject classifications

65T, 42B05