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An Optimal Ansatz Space for Moving Least Squares Approximation on Spheres

Ralf Hielscher, Tim Pöschl

Abstract

We revisit the moving least squares (MLS) approximation scheme on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields for functions in $\mathcal C^{L+1}(\mathbb S^{d-1})$ the approximation order $\mathcal O \left( h^{L+1} \right)$, where $h$ denotes the fill distance of the sampling nodes. In this paper we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degree up to $L$, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as $h \to 0$. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space as ansatz space.

An Optimal Ansatz Space for Moving Least Squares Approximation on Spheres

Abstract

We revisit the moving least squares (MLS) approximation scheme on the sphere , where . It is well known that using the spherical harmonics up to degree as ansatz space yields for functions in the approximation order , where denotes the fill distance of the sampling nodes. In this paper we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degree up to , while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as . Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space as ansatz space.
Paper Structure (6 sections, 14 theorems, 98 equations, 3 figures, 1 table)

This paper contains 6 sections, 14 theorems, 98 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Moving Least Squares Approximation
  3. A Minimal Ansatz Space for Local Approximation on Spheres
  4. A Uniform Bound for the Lebesgue Constant
  5. MLS Approximation with Polynomials on the Tangent Space of the Sphere
  6. Numerical Comparison

Key Result

Theorem 2.3

The MLS approximation can be written as where the coefficients $a_i^\ast (y) \in {\mathbb C}$ are the unique solution of the minimization problem

Figures (3)

  • Figure 1: Surface Plot of $(3+f(y)) \cdot y$
  • Figure 2: The fill distance $h$ and the separation distance $q$ in degree and the resulting uniformity $h/q$ for Fibonacci grids with $N = 2n+1$ nodes, $n \in \{ 5 \cdot 2^{1}, 5 \cdot 2^{2}, \dots, 5 \cdot 2^{26} \}$.
  • Figure :

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Backus-Gilbert Approximation
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 24 more