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Stable High-Order Interpolation on the Grassmann Manifold by Maximum-Volume Coordinates and Arnoldi Orthogonalization

Qiang Niu, Wen Jiang, Jie Fei, Ruoyu Xiong, Yuxuan Li

Abstract

High-order interpolation on the Grassmann manifold $\Gr(n, p)$ is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that combines Maximum-Volume (MV) local coordinates with Arnoldi-orthogonalized polynomial bases. First, manifold data are mapped to a well-conditioned Euclidean domain via MV coordinates. The approach bypasses the costly matrix factorizations inherent to traditional Riemannian normal coordinates. Within the coordinate space, we use the Vandermonde-with-Arnoldi (V+A) method for Lagrange interpolation and its confluent extension (CV+A) for derivative-enriched Hermite interpolation. By constructing discrete orthogonal bases directly from the parameter nodes, the solution of ill-conditioned linear system is avoided. Theoretical bounds are established to verify the stability of the geometric mapping and the polynomial approximation. Extensive numerical experiments demonstrate that the proposed MV-(C)V+A framework can produce highly accurate approximation in high-degree polynomial interpolation.

Stable High-Order Interpolation on the Grassmann Manifold by Maximum-Volume Coordinates and Arnoldi Orthogonalization

Abstract

High-order interpolation on the Grassmann manifold is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that combines Maximum-Volume (MV) local coordinates with Arnoldi-orthogonalized polynomial bases. First, manifold data are mapped to a well-conditioned Euclidean domain via MV coordinates. The approach bypasses the costly matrix factorizations inherent to traditional Riemannian normal coordinates. Within the coordinate space, we use the Vandermonde-with-Arnoldi (V+A) method for Lagrange interpolation and its confluent extension (CV+A) for derivative-enriched Hermite interpolation. By constructing discrete orthogonal bases directly from the parameter nodes, the solution of ill-conditioned linear system is avoided. Theoretical bounds are established to verify the stability of the geometric mapping and the polynomial approximation. Extensive numerical experiments demonstrate that the proposed MV-(C)V+A framework can produce highly accurate approximation in high-degree polynomial interpolation.
Paper Structure (15 sections, 7 theorems, 74 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 7 theorems, 74 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Interpolation on the Grassmann Manifold
  3. Vandermonde with Arnoldi and Its Confluent Extension
  4. The V+A Method
  5. The Confluent V+A Method
  6. The Integrated MV-(C)V+A Approach
  7. Stage I: Geometric Stabilization via Orthogonal Transformation
  8. Stage II: Algebraic Interpolation via Krylov Subspaces
  9. Stage III: Geometric Reconstruction
  10. Convergence and Error Analysis
  11. Error Propagation in Coordinate Systems
  12. Algebraic Stability of CV+A
  13. Total Error Bound
  14. Numerical Examples
  15. Conclusion

Key Result

Lemma 1

For any MV coordinates $\Xi \in \mathbb{R}^{(n-p) \times p}$, the reconstructed matrix satisfies $\widetilde{U}(\Xi)^T \widetilde{U}(\Xi) = I_p$, and its MV coordinates are exactly $\Xi$.

Figures (4)

  • Figure 1: Left: Relative reconstruction error $\|\hat{P}(t) - P(t)\|_F / \|P(t)\|_F$. Right: Orthogonality loss $\|\hat{U}(t)^T \hat{U}(t) - I_p\|_F$.
  • Figure 2: Left: Relative reconstruction error $\|\hat{P}(t) - P(t)\|_F / \|P(t)\|_F$. Right: Orthogonality loss $\|\hat{U}(t)^T \hat{U}(t) - I_p\|_F$.
  • Figure 3: Reconstruction error under data perturbation ($\epsilon = 10^{-10}$)
  • Figure 4: Relative POD subspace error $\| \hat{P}(t) - P_{\text{true}}(t) \|_F / \| P_{\text{true}}(t) \|_F$ for the parametric Helmholtz ROM. The global Hermite interpolation is of degree 23. Traditional methods relying on confluent Vandermonde systems suffer complete algebraic collapse, whereas MV-(C)V+A preserves machine precision.

Theorems & Definitions (17)

  • Remark 1
  • Definition 1
  • Lemma 1: Reconstruction and Orthogonality Preservation
  • proof
  • Lemma 2
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2: Optimal Algebraic Conditioning of CV+A
  • ...and 7 more