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Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

Michael Hecht, Krzysztof Gonciarz, Jannik Michelfeit, Vladimir Sivkin, Ivo F. Sbalzarini

TL;DR

This work advances multivariate polynomial interpolation by introducing non-tensorial unisolvent nodes that support Newton and Lagrange interpolation in arbitrary dimensions. By generalizing unisolvence and leveraging sub-exponential node growth with $A_{m,n,p}$ (notably $p=2$) while preserving exponential convergence (Trefethen rates) for analytic functions in the Trefethen domain, the authors lift much of the curse of dimensionality. They provide a practical, numerically stable algorithm with at most quadratic runtime (Newton) and linear memory, plus an efficient least-squares regression variant for scattered data. Empirical results on Runge-type functions up to dimension five validate the theoretical rates and demonstrate competitiveness against traditional tensor-grid methods, with additional discussion on extensions to manifolds, barycentric forms, and regression tasks across curved domains.

Abstract

We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

TL;DR

This work advances multivariate polynomial interpolation by introducing non-tensorial unisolvent nodes that support Newton and Lagrange interpolation in arbitrary dimensions. By generalizing unisolvence and leveraging sub-exponential node growth with (notably ) while preserving exponential convergence (Trefethen rates) for analytic functions in the Trefethen domain, the authors lift much of the curse of dimensionality. They provide a practical, numerically stable algorithm with at most quadratic runtime (Newton) and linear memory, plus an efficient least-squares regression variant for scattered data. Empirical results on Runge-type functions up to dimension five validate the theoretical rates and demonstrate competitiveness against traditional tensor-grid methods, with additional discussion on extensions to manifolds, barycentric forms, and regression tasks across curved domains.

Abstract

We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

Paper Structure

This paper contains 29 sections, 16 theorems, 75 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Runge's phenomenon -- Approximation theory
  3. Lifting the curse of dimensionality
  4. Interpolation with optimal Trefethen rates -- The core contribution of this article
  5. Related Work
  6. Interpolation in tensorial (sub) grids
  7. Weierstrass-type Approximation
  8. Multivariate splines
  9. Notation
  10. Unisolvent nodes
  11. The notion of unisolvence
  12. Construction of unisolvent nodes
  13. Multivariate Newton interpolation
  14. Multivariate Lagrange interpolation
  15. Regression on scattered data
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $f : \Omega \subseteq \mathbb{R}^2 \longrightarrow \mathbb{R}$ be a $(n+1)$-times continuously differentiable bivariate function, $\Delta$ be a triangulation, and $S_{f,n,\Delta} = \left\{g \in C^\rho(\Omega,\mathbb{R}) \,\,|\,\, g_{|\delta} \in \Pi_{m,n,1}\,, \,\, \forall\, \delta \in \Delta \r where $|\Delta| = \max_{T \in \Delta} |T|$ is the mesh size.

Figures (9)

  • Figure 1: Examples of unisolvent nodes $P_A$ for $A= A_{2,3,1}$ in general (left) and on (irregular) grids (middle, right). Non-tensorial nodes are indicated in red with missing symmetric counterparts shown as open symbols (right).
  • Figure 2: Recursive interpolation using the generalized divided difference scheme from Corollary \ref{['cor:GDDS']}.
  • 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 as well as non-tensorial nodes in red with missing symmetric counterparts shown as open symbols.
  • Figure 4: Unisolvent nodes $P_A$ in 2D (left) and 3D (right) with respect to $A_{m,n,p}$ for dimensions $m=2,3$, $n=5$, $p=2$, and Leja ordered leja generating nodes $\mathrm{GP} = \oplus_{i=1}^m(-1)^m\mathrm{Cheb}_n^{0}$. Nodes belonging to the same line/plane are colored equally.
  • Figure 5: Approximation errors for the benchmarked methods interpolating the Runge function in dimension $m=2$.
  • ...and 4 more figures

Theorems & Definitions (52)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1: Carl de Boor
  • Definition 2.1: Transformations
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Unisolvent nodes
  • ...and 42 more