Bravais Lattices for Euclidean Degree Efficient Polynomial Interpolation

TL;DR

This paper introduces a new polynomial interpolation method on squares and cubes that is more efficient in Euclidean degree, leveraging FFT for evaluation, with better accuracy and stability than traditional tensor product methods.

Contribution

The paper presents a novel Euclidean degree-based polynomial interpolation approach with improved efficiency, accuracy, and stability, extending prior total degree methods.

Findings

Efficient interpolation and evaluation via FFT.

Minimally growing Lebesgue constant.

Outperforms tensor product Gauss-Legendre integration.

Abstract

A method is presented for forming polynomial interpolants on squares and cubes, which are more efficient in the so-called Euclidean degree than other commonly used methods with the same number of collocation points. These methods have several additional desirable properties. The interpolants can be formed and evaluated via the FFT and have a minimally growing Lebesgue constant. The associated points achieve Gauss-Lobatto order accuracy in integration, out-performing tensor product Gauss-Legendre integration for many $C_\infty$ functions. This method is related to prior work on total degree efficient collocation points by Yuan Xu et al. [arXiv:math/0604604] [arXiv:0808:0180]