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The Unisolvence of Lagrange Interpolation with Symmetric Interpolation Space and Nodes in High Dimension

Yulin Xie, Yifa Tang

TL;DR

This work addresses unisolvence in high-dimensional Lagrange interpolation under symmetry constraints. By embedding the problem in group action and permutation representation theory, it derives that when the interpolation space and node set are symmetric under $S_n$, the node symmetry is uniquely determined by the space, with the Gram-like matrix $V$ encoding the relation between basis and nodes. The authors prove that $V$ is a symmetric positive definite Gram matrix and invertible, which ensures a unique correspondence between orbit structures of basis functions and nodes, generalizing conjectures and yielding practical guidance for selecting nodes. The results extend beyond polynomial interpolation to general interpolation problems, and they outline a path toward a complete necessary-and-sufficient condition for unisolvence in symmetric settings. Practical impact lies in reducing the search space for symmetric nodes in high dimensions while preserving unisolvence.

Abstract

High-dimensional Lagrange interpolation plays a pivotal role in finite element methods, where ensuring the unisolvence and symmetry of its interpolation space and nodes set is crucial. In this paper, we leverage group action and group representation theories to precisely delineate the conditions for unisolvence. We establish a necessary condition for unisolvence: the symmetry of the interpolation nodes set is determined by the given interpolation space. Our findings not only contribute to a deeper theoretical understanding but also promise practical benefits by reducing the computational overhead associated with identifying appropriate interpolation nodes.

The Unisolvence of Lagrange Interpolation with Symmetric Interpolation Space and Nodes in High Dimension

TL;DR

This work addresses unisolvence in high-dimensional Lagrange interpolation under symmetry constraints. By embedding the problem in group action and permutation representation theory, it derives that when the interpolation space and node set are symmetric under , the node symmetry is uniquely determined by the space, with the Gram-like matrix encoding the relation between basis and nodes. The authors prove that is a symmetric positive definite Gram matrix and invertible, which ensures a unique correspondence between orbit structures of basis functions and nodes, generalizing conjectures and yielding practical guidance for selecting nodes. The results extend beyond polynomial interpolation to general interpolation problems, and they outline a path toward a complete necessary-and-sufficient condition for unisolvence in symmetric settings. Practical impact lies in reducing the search space for symmetric nodes in high dimensions while preserving unisolvence.

Abstract

High-dimensional Lagrange interpolation plays a pivotal role in finite element methods, where ensuring the unisolvence and symmetry of its interpolation space and nodes set is crucial. In this paper, we leverage group action and group representation theories to precisely delineate the conditions for unisolvence. We establish a necessary condition for unisolvence: the symmetry of the interpolation nodes set is determined by the given interpolation space. Our findings not only contribute to a deeper theoretical understanding but also promise practical benefits by reducing the computational overhead associated with identifying appropriate interpolation nodes.
Paper Structure (19 sections, 15 theorems, 22 equations, 1 table)

This paper contains 19 sections, 15 theorems, 22 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A group acts on a set
  4. Group representation
  5. The symmetry and interpolation in $\mathbb{R}^n$
  6. The unisolvence of Lagrange interpolation in $\mathbb{R}^n$
  7. $S_n$ acts on $\mathbb{R}^n$
  8. $S_n$ acts on $f:\mathbb{R}^n\to\mathbb{R}$
  9. Symmetric Lagrange interpolation
  10. A necessary condition for nodes set
  11. The type and equivalence of orbits in $\mathbb{R}^n$
  12. The equivalence between finite symmetric subsets
  13. unisolvence and group action
  14. Symmetry and reversibility of matrix V
  15. $V$ is symmetric
  16. ...and 4 more sections

Key Result

Lemma 2.7

For a finite group $G$ that acts on a finite set $X$, let $t$ be the number of orbits. Then, the Burnside's Lemma states that The number of orbits is equal to the average of number of fixed points of elements of $G$.

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7: Burnside's Lemma
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 41 more