Generalized local polynomial reproductions

TL;DR

This work develops a coordinate-free, mesh-free approximation framework on Lipschitz domains within Riemannian manifolds, proving the existence of norming sets and generalized local polynomial reproductions that enable stable, local representations of polynomials. It then constructs moving least squares (MLS) on manifolds without relying on tangent-plane projections, deriving stability, regularity, and locality properties of the MLS shape functions and their derivatives. The theory is applied to algebraic manifolds to produce smooth, local polynomial reproductions with rigorous error and stability guarantees for RBF interpolation and MLS approximations, including noisy data scenarios. Numerical experiments on cyclide-like manifolds and meshed surfaces validate the locality and convergence rates, demonstrating the practical viability and robustness of coordinate-free MLS for mesh-free approximation on manifolds.

Abstract

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of various mesh-free methods and a mesh-free method in its own right. As a key application, we prove the existence of smooth local polynomial reproductions on compact subsets of algebraic manifolds in $\mathbb{R}^n$ with Lipschitz boundary. These results are then applied to derive new findings on the existence, stability, regularity, locality, and approximation properties of shape functions for a coordinate-free moving least squares approximation method on algebraic manifolds, which operates directly on point clouds without requiring tangent plane approximations. There are two appendices: the first derives high order Markov inequalities for polynomials on algebraic manifolds and the second gives instructions for calculating the dimension of the space of degree $m$ polynomials restricted to a real algebraic variety.

Figures (7)

• Figure 1: A diagram indicating the points $x_0$, $z$ and $\zeta$.
• Figure 2: Left: Distribution of centers $\Xi$ on the torus. Middle and right: heat map of the power function $P_{\Xi}$ (displayed on a $\log_{10}$ scale) using the Matérn kernel with $s=4$ and $s=5$, respectively.
• Figure 3: Left panel: visualization of the manifold $\mathbb{M}$ and the compact subset $\widehat{\Omega}\subset\mathbb{M}$ (inset) used in the MLS numerical experiments; see \ref{['eq:cyclide']} and \ref{['eq:patch']} for exact definitions. Black solid spheres mark the node sets $\Xi$ on $\mathbb{M}$ and $\widehat{\Xi}$ on $\widehat{\Omega}$. Right panel: heat map of the target function on $\mathbb{M}$ considered in the numerical experiments; the brighter region highlights the target function on $\widehat{\Omega}$.
• Figure 4: Convergence results where the approximation problem is done over (a) all of $\mathbb{M}$ and (b) only the compact subset $\widehat{\Omega}$. The estimated rates of convergence for each $m$ are included in legend labels.
• Figure 5: Numerically computed Lebesgue constants for the MLS approximation problem for the compact subset $\widehat{\Omega}$ of cyclide using quasi-uniform points of increasing cardinality $N_{\widehat{\Xi}}$.

Theorems & Definitions (43)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.3.1: Norming Set
• proof
• Corollary 3.3.2: Nikolskii
• Lemma 4.1