Multivariate Data-dependent Partition of Unity based on Moving Least Squares method

TL;DR

This paper introduces a novel data-dependent partition of unity method based on Moving Least Squares, enhancing accuracy near discontinuities in multivariate data approximation while preserving high-order accuracy in smooth regions.

Contribution

It extends MLS with a new Non-Linear Partition of Unity approach for higher dimensions, improving handling of discontinuities and maintaining accuracy.

Findings

Improved accuracy near discontinuities.

Maintains high-order accuracy in smooth regions.

Validated effectiveness through numerical experiments.

Abstract

Data approximation is essential in fields such as geometric design, numerical PDEs, and curve modeling. Moving Least Squares (MLS) is a widely used method for data fitting; however, its accuracy degrades in the presence of discontinuities, often resulting in spurious oscillations similar to those associated with the Gibbs phenomenon. This work extends the integration of MLS with the Weighted Essentially Non-Oscillatory (WENO) method and with an innovative partition of unity approach to higher dimensions. We propose a data-dependent operator using the novel Non-Linear Partition of Unity based on Moving Least Squares method in $\mathbb{R}^n$, which improves accuracy near discontinuities and maintains high-order accuracy in smooth regions. We demonstrate some theoretical properties of the method and perform numerical experiments to validate its effectiveness.