Transformed rank-1 lattices for high-dimensional approximation

TL;DR

This paper extends Fourier approximation techniques for multivariate functions on the torus to weighted spaces via a change of variables, using transformed rank-1 lattices and dimension-incremental methods, with numerical validation.

Contribution

It introduces a new approach combining transformations and rank-1 lattices for high-dimensional Fourier approximation in weighted spaces, with theoretical and numerical validation.

Findings

The method achieves accurate approximation for high-dimensional functions.

Numerical tests confirm the theoretical error bounds.

Dimension-incremental methods effectively handle sparse frequency sets.

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.