Approximation of multivariate periodic functions based on sampling along multiple rank-1 lattices

TL;DR

This paper develops a sampling method using multiple rank-1 lattices for reconstructing high-dimensional periodic functions, achieving near-optimal error rates and demonstrating high efficiency through theoretical analysis and numerical tests.

Contribution

It introduces the use of reconstructing multiple rank-1 lattices for high-dimensional function approximation, providing error estimates and confirming their effectiveness through numerical experiments.

Findings

Error rates close to optimal for Sobolev spaces

High reliability and low oversampling in sampling schemes

Numerical tests confirm theoretical predictions

Abstract

In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable properties such as easy constructability, the existence of high-dimensional fast Fourier transform algorithms, high reliability, and low oversampling factors. Especially, we show error estimates for functions from Sobolev Hilbert spaces of generalized mixed smoothness. For instance, when measuring the sampling error in the $L_2$-norm, we show sampling error estimates where the exponent of the main part reaches those of the optimal sampling rate except for an offset of $1/2+\varepsilon$, i.e., the exponent is almost a factor of two better up to the mentioned offset compared to single rank-1 lattice sampling. Various numerical tests in medium and high dimensions demonstrate the high performance and confirm the obtained theoretical results of multiple rank-1 lattice sampling.