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Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D

Inmaculada Garcés, José M. Ramón, Juan Ruiz-Álvarez, Dionisio F. Yáñez

TL;DR

The paper tackles Gibbs-like artifacts in Moving Least Squares (MLS) when data contain discontinuities by integrating MLS with a non-linear Weighted Essentially Non-Oscillatory (WENO) approach within a partition-of-unity framework in 1D. It develops a nonlinear operator $\mathcal{Q}^{NL}_{PU}$ that preserves $O(h^{d+1})$ accuracy in smooth regions while adaptively downweights non-smooth stencils near discontinuities, leveraging smoothness indicators $\mathcal{I}_k$ and WENO-like weights. Theoretical results establish polynomial reproduction for PV MLS, and, under suitable conditions, $O(h^{d+1})$ convergence in smooth parts; numerical experiments corroborate high-order accuracy and effective suppression of oscillations near jumps. The method broadens MLS applicability to discontinuous data and suggests pathways to extensions in multi-dimensions and multiresolution/imagery applications.

Abstract

The approximation of data is a fundamental challenge encountered in various fields, including computer-aided geometric design, the numerical solution of partial differential equations, or the design of curves and surfaces. Numerous methods have been developed to address this issue, providing good results when the data is continuous. Among these, the Moving Least Squares (MLS) method has proven to be an effective strategy for fitting data, finding applications in both statistics and applied mathematics. However, the presence of isolated discontinuities in the data can lead to undesirable artifacts, such as the Gibbs phenomenon, which adversely affects the quality of the approximation. In this paper, we propose a novel approach that integrates the Moving Least Squares method with the well-established non-linear Weighted Essentially Non-Oscillatory (WENO) method. This combination aims to construct a non-linear operator that enhances the accuracy of approximations near discontinuities while maintaining the order of accuracy in smooth regions. We investigate the properties of this operator in one dimension, demonstrating its effectiveness through a series of numerical experiments that validate our theoretical findings.

Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D

TL;DR

The paper tackles Gibbs-like artifacts in Moving Least Squares (MLS) when data contain discontinuities by integrating MLS with a non-linear Weighted Essentially Non-Oscillatory (WENO) approach within a partition-of-unity framework in 1D. It develops a nonlinear operator that preserves accuracy in smooth regions while adaptively downweights non-smooth stencils near discontinuities, leveraging smoothness indicators and WENO-like weights. Theoretical results establish polynomial reproduction for PV MLS, and, under suitable conditions, convergence in smooth parts; numerical experiments corroborate high-order accuracy and effective suppression of oscillations near jumps. The method broadens MLS applicability to discontinuous data and suggests pathways to extensions in multi-dimensions and multiresolution/imagery applications.

Abstract

The approximation of data is a fundamental challenge encountered in various fields, including computer-aided geometric design, the numerical solution of partial differential equations, or the design of curves and surfaces. Numerous methods have been developed to address this issue, providing good results when the data is continuous. Among these, the Moving Least Squares (MLS) method has proven to be an effective strategy for fitting data, finding applications in both statistics and applied mathematics. However, the presence of isolated discontinuities in the data can lead to undesirable artifacts, such as the Gibbs phenomenon, which adversely affects the quality of the approximation. In this paper, we propose a novel approach that integrates the Moving Least Squares method with the well-established non-linear Weighted Essentially Non-Oscillatory (WENO) method. This combination aims to construct a non-linear operator that enhances the accuracy of approximations near discontinuities while maintaining the order of accuracy in smooth regions. We investigate the properties of this operator in one dimension, demonstrating its effectiveness through a series of numerical experiments that validate our theoretical findings.
Paper Structure (11 sections, 4 theorems, 53 equations, 3 figures, 3 tables)

This paper contains 11 sections, 4 theorems, 53 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. MLS for point-value (PV) data
  3. WENO method
  4. Outline of this paper
  5. Reviewing the general solution of MLS problem
  6. Point-value MLS in 1d
  7. WENO-like MLS based on the partition of unity method
  8. Numerical experiments
  9. Order of accuracy
  10. Avoiding oscillations close to the discontinuities
  11. Conclusions and future work

Key Result

Proposition 2.1

With the same notation as before, $C_{i_0}$ has the explicit form

Figures (3)

  • Figure 1: Approximation to the function $g$ (black), Eq. \ref{['funciong']}, using linear and non-linear methods with ${\text{W2}}$ (red), ${\text{W4}}$ (blue) and G (magenta).
  • Figure 2: Zoom around the discontinuity of the approximation to the function $g$ (black), Eq. \ref{['funciong']}, using the linear and non-linear methods with ${\text{W2}}$ (red), ${\text{W4}}$ (blue) and G (magenta).
  • Figure 3: Approximation to function $z$ (black), Eq. \ref{['funcionh']}, using linear and non-linear methods with ${\text{W2}}$ (red) and ${\text{W4}}$ (blue) and G (magenta); $p=3$.

Theorems & Definitions (6)

  • Definition 1: The MLS approximation
  • Proposition 2.1
  • Example 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 3.1