Data dependent Moving Least Squares

TL;DR

This paper introduces a data-dependent modification to the moving least squares method that reduces Gibbs phenomenon near discontinuities by using smoothness indicators to adjust weights, improving approximation accuracy.

Contribution

It proposes a novel weight function based on data smoothness, enhancing MLS by mitigating Gibbs phenomenon and improving discontinuity representation.

Findings

Reduced Gibbs phenomenon near discontinuities

Improved approximation accuracy in data with discontinuities

Validated through numerical experiments

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.