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Bravais Lattices for Euclidean Degree Efficient Polynomial Interpolation

R. Connor Greene

TL;DR

The work develops an efficient Euclidean-degree framework for multivariate Chebyshev interpolation on $[-1,1]^d$, leveraging lattice-based point sets and FFT-enabled transforms. By formulating both Bravais and composite lattices and introducing IlFFT and HFFT algorithms, the method achieves $O(n\log n)$-type scaling while reducing point counts relative to tensor-product grids. Across 2D and 3D, the approach demonstrates competitive or superior integration and interpolation accuracy compared with Gauss-Legendre and Clenshaw-Curtis stencils, particularly when the Euclidean-degree basis is well matched to the lattice. The results suggest practical gains in computation time and accuracy for high-dimensional polynomial approximation, with clear avenues for further optimization and extension to 3D composites and improved low-resolution behavior.

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]

Bravais Lattices for Euclidean Degree Efficient Polynomial Interpolation

TL;DR

The work develops an efficient Euclidean-degree framework for multivariate Chebyshev interpolation on , leveraging lattice-based point sets and FFT-enabled transforms. By formulating both Bravais and composite lattices and introducing IlFFT and HFFT algorithms, the method achieves -type scaling while reducing point counts relative to tensor-product grids. Across 2D and 3D, the approach demonstrates competitive or superior integration and interpolation accuracy compared with Gauss-Legendre and Clenshaw-Curtis stencils, particularly when the Euclidean-degree basis is well matched to the lattice. The results suggest practical gains in computation time and accuracy for high-dimensional polynomial approximation, with clear avenues for further optimization and extension to 3D composites and improved low-resolution behavior.

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 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]

Paper Structure

This paper contains 26 sections, 1 theorem, 16 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Map from the Circle to the Interval
  4. Chebyshev Polynomials
  5. Discrete Cosine Transform
  6. Chebyshev Polynomials in Higher Dimensions
  7. Euclidean Degree Efficiency
  8. Lattices and Aliasing
  9. Special Properties of Chebyshev Lattices
  10. Composite Lattices
  11. Optimal Lattices in Two and Three Dimensions
  12. 2d
  13. 3d
  14. Higher Dimensions
  15. Composite Lattices
  16. ...and 11 more sections

Key Result

Theorem 1

The following statements are true:

Figures (10)

  • Figure 1: 11th order Chebyshev points of the Second kind mapped on the interval, and a semi-circular path. Note the uniform distribution of the nodes on the semi-circle.
  • Figure 2: From left to right: example point distributions for DCT I, II, $\mathrm{V}_-$, and $\mathrm{V}_+$.
  • Figure 3: Reciprocal lattices to the Padua points and Hex points respectively. Note the circle representing the Euclidean degree and the red dots representing aliasing frequencies.
  • Figure 4: Left: Example of a seven point composite lattice. The dotted box contains a single unit cell. Right: An example index set of the basis for a seven-point composite lattice. Coefficients whose indices are plotted with matching colors are stored in a single contiguous storage array.
  • Figure 5: From left to right: Cartesian, Hexagonal, and Padua lattices of Euclidean degree 8. The point counts are 81, 68, and 78 respectively.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2: Bravais Lattice
  • Theorem 1
  • proof
  • Definition 3: Composite Lattice
  • Definition 4: Brillouin Zone