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Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions

Michael Hecht, Phil-Alexander Hofmann, Damar Wicaksono, Uwe Hernandez Acosta, Krzysztof Gonciarz, Jannik Kissinger, Vladimir Sivkin, Ivo F. Sbalzarini

TL;DR

This paper extends univariate Newton interpolation to arbitrary dimensions for downward-closed polynomial spaces, achieving quadratic runtime and linear storage while accommodating non-tensorial unisolvent node grids. It introduces BLT-function theory to establish geometric (exponential) convergence rates that extend to derivatives, with Leja-ordered nodes (notably Leja-ordered Chebyshev–Lobatto) yielding near-optimal rates and mitigating the curse of dimensionality via Euclidean degree. The authors develop a multivariate Newton interpolation (MIP) algorithm that works with arbitrary downward-closed spaces, provides efficient evaluation and differentiation, and proves rate results via Lebesgue-constant analyses; they also implement and benchmark the approach in the minterpy package, demonstrating strong empirical performance against state-of-the-art methods. The practical impact is a scalable, stable interpolation framework suitable for high-dimensional spectral-type tasks in PDEs, ODEs, and signal processing, with open-source tools enabling broad adoption and further optimization toward $\mathcal{O}(N m n)$ time.

Abstract

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions

TL;DR

This paper extends univariate Newton interpolation to arbitrary dimensions for downward-closed polynomial spaces, achieving quadratic runtime and linear storage while accommodating non-tensorial unisolvent node grids. It introduces BLT-function theory to establish geometric (exponential) convergence rates that extend to derivatives, with Leja-ordered nodes (notably Leja-ordered Chebyshev–Lobatto) yielding near-optimal rates and mitigating the curse of dimensionality via Euclidean degree. The authors develop a multivariate Newton interpolation (MIP) algorithm that works with arbitrary downward-closed spaces, provides efficient evaluation and differentiation, and proves rate results via Lebesgue-constant analyses; they also implement and benchmark the approach in the minterpy package, demonstrating strong empirical performance against state-of-the-art methods. The practical impact is a scalable, stable interpolation framework suitable for high-dimensional spectral-type tasks in PDEs, ODEs, and signal processing, with open-source tools enabling broad adoption and further optimization toward time.

Abstract

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

Paper Structure

This paper contains 20 sections, 11 theorems, 37 equations, 11 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Bos--Levenberg--Trefethen functions
  3. Related work and contribution
  4. Notation
  5. The notion of unisolvence
  6. Unisolvent nodes
  7. Unisolvent non-tensorial grids
  8. Multivariate Newton interpolation in non-tensorial grids
  9. Multivariate Newton interpolation
  10. Multivariate Lagrange interpolation
  11. Leja-ordered nodes and Lebesgue constants
  12. Numerical experiments
  13. MIP benchmarks
  14. minterpy benchmarks
  15. Discussion of results for \ref{['F1']}
  16. ...and 5 more sections

Key Result

Theorem 1

Given a continuous function $f : \square_m \longrightarrow \mathbbm{R}$ satisfying Eq. eq:cheb, assume that $f$ possesses an analytic (holomorphic) extension to the Trefethen domain where $E_{m,h^2}^2$ denotes the Newton ellipse with foci $0$ and $m$ and leftmost point $-h^2$, $h \in [0,1]$. Setting $\rho = h + \sqrt{1 + h^2}$, the following upper bounds on the convergence rate of the truncation $

Figures (11)

  • Figure 1: Examples of unisolvent nodes $P_A$ for $A= A_{2,3,1}$ in general (left), irregular (middle), and non-tensorial (right) grids. In the right panel, non-tensorial nodes are indicated in red with missing symmetric counterparts shown as open symbols.
  • Figure 2: The generalised divided difference scheme given by recursively choosing suitable hyper(sub)planes $H_{0}, H_{1,0}, H_{0,1}, \ldots$ and nodes $P=P_0 =P_{1,0}\cup P_{0,1}, \ldots$ according to Theorem \ref{['theorem:UN']}, and applying the splitting Theorem \ref{['GS']} to the separated polynomials $Q_{1},Q_2,Q_{1,0},Q_{2,0},Q_{1,1},Q_{2,1},\ldots$.
  • Figure 3: Examples of unisolvent nodes for $A= A_{2,3,1}$ (left, middle) and $A_{2,3,2}$ (right). Note that $(2,2) \in A_{2,3,2}\setminus A_{2,3,1}$ generates an extra node. Orderings in $x,y$--directions are indicated by numbers, and non-tensorial nodes are shown in red.
  • Figure 4: Leja-ordered Chebyshev-Lobatto (LCL) nodes $P_A$, $A=A_{m,5,2}$ in 2D ($m=2$, left) and 3D ($m=3$, right). Nodes in the same horizontal hyperplane are coloured equally.
  • Figure 5: Numerically measured Lebesgue constants for LCL nodes (solid lines) and LP nodes (dashed lines) $P_A \subseteq \square_m$ for total ($p=1$, green), Euclidean ($p=2$, red), and maximum degree ($p=\infty$, blue) polynomial interpolants in dimensions $m =1,\dots,4$ (panels from left to right).
  • ...and 6 more figures

Theorems & Definitions (36)

  • Theorem 1: trefethen2017a
  • Theorem 2: bos2018bernstein
  • Definition 1: Bos--Levenberg--Trefethen functions
  • Remark 1
  • Definition 2: Transformations
  • Lemma 1
  • proof
  • Definition 3
  • Example 1
  • Definition 4: Unisolvence for hyperplane splits
  • ...and 26 more