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$\mathcal{H}$-inverses for RBF interpolation

Niklas Angleitner, Markus Faustmann, Jens Markus Melenk

TL;DR

This paper establishes that for RBF interpolation with kernels including polyharmonic splines, the inverse interpolation matrix admits an $\mathcal{H}$-matrix approximation with exponential convergence in block rank when the block partition follows standard clustering. The authors develop a function-level reduction via a native space framework, define localized harmonic spaces $V_{\mathrm{harm}}(B)$, and construct low-rank, localized operators (cut-off, $\Pi_H$, and coarsening) to approximate the inverse blocks. The key theoretical contribution is a bound of the form $\|\boldsymbol{S_{11}}-\boldsymbol{M}\|_2 \lesssim \ln(N) N^{\sigma_{\mathrm{card}}(3k-d)/d} \exp(-\sigma_{\mathrm{exp}} r^{1/(d+1)})$, underpinning log-linear storage and fast arithmetic in $\mathcal{H}$-matrix format. Numerical experiments in 2D/3D confirm the exponential decay of block approximations and illustrate practical gains for large nonuniform point clouds and various kernels, including Matérn-like and Bessel-potential variants.

Abstract

We consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the $\mathcal{H}$-matrix format if the block structure of the $\mathcal{H}$-matrix arises from a standard clustering algorithm.

$\mathcal{H}$-inverses for RBF interpolation

TL;DR

This paper establishes that for RBF interpolation with kernels including polyharmonic splines, the inverse interpolation matrix admits an -matrix approximation with exponential convergence in block rank when the block partition follows standard clustering. The authors develop a function-level reduction via a native space framework, define localized harmonic spaces , and construct low-rank, localized operators (cut-off, , and coarsening) to approximate the inverse blocks. The key theoretical contribution is a bound of the form , underpinning log-linear storage and fast arithmetic in -matrix format. Numerical experiments in 2D/3D confirm the exponential decay of block approximations and illustrate practical gains for large nonuniform point clouds and various kernels, including Matérn-like and Bessel-potential variants.

Abstract

We consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the -matrix format if the block structure of the -matrix arises from a standard clustering algorithm.

Paper Structure

This paper contains 20 sections, 28 theorems, 103 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The interpolation problem
  4. The fundamental solution
  5. The LSE
  6. Hierarchical matrices
  7. The main result
  8. Proof of main results
  9. The native space $V$
  10. An equivalent interpolation problem
  11. Properties of the LSE
  12. The representation formula for the inverse matrix
  13. Reduction from matrix-level to function-level
  14. The cut-off operator
  15. The space $V_{\mathrm{harm}}(B)$
  16. ...and 5 more sections

Key Result

Lemma 2.5

If $k_{\min} = 0$, then there exists a unique function $\varphi \in V$, such that the following equality holds true:

Figures (5)

  • Figure 1: Interpolation of smooth data on a non-uniform point distribution.
  • Figure 2: A "typical" hierarchical block partition and a "typical" error plot in $2$D.
  • Figure 3: A comparison of different problem sizes $N$ for a uniform 3D grid.
  • Figure 4: Experimenting with an algebraically graded grid in 3D.
  • Figure 5: Experiment using $\mathcal{H}$-arithmetics to approximate the inverse system matrix, left: 2D thin-plate splines, right: 3D Bessel potential.

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.10
  • ...and 63 more