## 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 $\psi:\left(-\frac{1}{2},\frac{1}{2}\right)^d\to\mathbb{R}^d$. 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 $H_{\rm mix}^{m}(\mathbb{T}^d)$ 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 the obtained theoretical results for the transformed methods.

Full Text (PDF) [1.3 MB], BibTeX

### Key words

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

65T, 42B05

< Back