Data dependent Moving Least Squares

TL;DR

The paper introduces a data-dependent modification of Moving Least Squares (MLS) by replacing standard distance-based weights with $\tilde{\omega}_i(\mathbf{x})=\frac{\omega_i(\mathbf{x})}{(\epsilon+I_i)^t}$, where $I_i$ are smoothness indicators inspired by WENO. This approach identifies nodes near discontinuities (infected nodes) and reduces their influence, mitigating Gibbs oscillations and over-smearing near jumps while preserving polynomial reproduction and achieving $O(h^{d+1})$ accuracy for functions in $C^{d+1}$. Theoretical results prove smoothness, polynomial reproduction, and error bounds, along with a reduced diffusion region near discontinuities. Numerical experiments across smooth and discontinuous test data, different point distributions, and weight choices confirm reduced oscillations and diffusion, validating the method's effectiveness and suggesting directions for extending the framework to other data-structure criteria.

Abstract

In this paper, we address a data dependent modification of the moving least squares (MLS) problem. We propose a novel approach by replacing the traditional weight functions with new functions that assign smaller weights to nodes that are close to discontinuities, while still assigning smaller weights to nodes that are far from the point of approximation. Through this adjustment, we are able to mitigate the undesirable Gibbs phenomenon that appears close to the discontinuities in the classical MLS approach, and reduce the smearing of discontinuities in the final approximation of the original data. The core of our method involves accurately identifying those nodes affected by the presence of discontinuities using smoothness indicators, a concept derived from the data-dependent WENO method. Our formulation results in a data-dependent weighted least squares problem where the weights depend on two factors: the distances between nodes and the point of approximation, and the smoothness of the data in a region of predetermined radius around the nodes. We explore the design of the new data-dependent approximant, analyze its properties including polynomial reproduction, accuracy, and smoothness, and study its impact on diffusion and the Gibbs phenomenon. Numerical experiments are conducted to validate the theoretical findings, and we conclude with some insights and potential directions for future research.

Figures (10)

• Figure 1: Approximation to the function $f$, Eq. \ref{['funcion1']}, where $f_1$ is the Franke's function, $f_2$ is the Franke's function plus a constant, and the discontinuity curve is defined by the zero level of the level-set function $\gamma(x,y)=0.25^2-x^2-y^2$. We have used the MLS with $\omega(x)$ the $\mathcal{C}^2$ Wendland function wendland2002 and the class of polynomials $\Pi_2(\mathbb{R}^2)$. Red points: original function, blue lines: approximation.
• Figure 2: Approximation to the function $g$, Eq. \ref{['ejemplolevin']}, using linear and data dependent MLS with different $\omega(x)$ functions wendland2002 and the class of polynomials $\Pi_2(\mathbb{R}^2)$. Red points: original function, blue lines: approximation.
• Figure 3: Approximation to the function $z$, Eq. \ref{['ejemplonuma']}, using linear and data dependent MLS with $\omega(x)$ the $\mathcal{C}^2$ Wendland function wendland2002 and the class of polynomials $\Pi_2(\mathbb{R}^2)$. Red points: original function, blue lines: approximation.
• Figure 4: Approximation to the function $z$, Eq. \ref{['ejemplonuma']}, using linear and data dependent MLS with $\omega(x)$ the $\mathcal{C}^\infty$ Gauss function wendland2002 and the class of polynomials $\Pi_2(\mathbb{R}^2)$. Red points: original function, blue lines: approximation.
• Figure 5: Approximation to function $z$, Eq. \ref{['ejemplonuma']}, using W2, $d=0$ and $N=33^2$.

Theorems & Definitions (5)

• Remark 1
• Theorem 3.4
• Remark 2
• Theorem 3.5
• Corollary 3.6