# Approximation of high-dimensional periodic functions with Fourier-based methods

## TL;DR

This paper introduces a Fourier-based approximation method for high-dimensional periodic functions using multivariate ANOVA decomposition, emphasizing sparsity and efficient algorithms for scattered data and black-box scenarios.

## Contribution

It develops a novel high-dimensional approximation technique leveraging ANOVA decomposition, smoothness inheritance, and specialized Fourier algorithms for improved efficiency.

## Key findings

- Effective importance ranking of dimensions and interactions.
- Utilization of NFFT for fast Fourier matrix multiplication.
- Properties of rank-1 lattices for black-box approximation.

## Abstract

In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important properties such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim to approximate a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimension interactions which is referred to as attribute ranking in some applications. In scattered data approximation we rely on a special algorithm based on the non-equispaced fast Fourier transform (or NFFT) for fast multiplication with arising Fourier matrices. For black-box approximation we choose the well-known rank-1 lattices as sampling schemes and show properties of the appearing special lattices.