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

Inmaculada Garcés, Juan Ruiz-Álvarez, Dionisio F. Yáñez

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.

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

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 , 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.
Paper Structure (10 sections, 7 theorems, 58 equations, 10 figures, 5 tables)

This paper contains 10 sections, 7 theorems, 58 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The MLS method
  3. The PU method
  4. The RBF method
  5. Integrating MLS with the Partition of Unity method in $\mathbb{R}^n$
  6. The DDPU-MLS method
  7. Numerical experiments
  8. Order of accuracy
  9. Avoiding oscillations close to the discontinuities
  10. Conclusions

Key Result

Theorem 2.1

Suppose that $\Omega \subseteq \mathbb{R}^n$ is bounded. Let us define $\widetilde{\Omega}$ as the closure of $\bigcup_{x \in \Omega} B(x, C_2 h_0)$. Assume that $f \in C^{m+1}(\widetilde{\Omega})$. Let us define $\mathcal{Q}(f)(\mathbf{x}) = \sum_{i=1}^N a_i f(\mathbf{x}_i)$, where $\{a_i\}_{i=1}^ for all $X$ with $h_{X,\Omega} \le h_0$. The semi-norm appearing on the right-hand side is given by

Figures (10)

  • Figure 1: (a) MLS approximation of Franke’s function given in Eq. \ref{['Franke']}; (b) MLS approximation of the function ${\color{black} f_2}$ defined in Eq. \ref{['frank']}; (c) Rotated view of plot (b).
  • Figure 2: Ball in a cone.
  • Figure 3: Approximation of the function $f_2$, Eq. \ref{['frank']}, using PU-MLS and DDPU-MLS with the Wendland $\mathcal{C}^2$ function. The second column shows rotated views of the plots in the first column. The third column shows the errors between the original function and its approximation.
  • Figure 4: Approximation of the function $f_2$, Eq. \ref{['frank']}, using PU-MLS and DDPU-MLS with the Wendland $\mathcal{C}^4$ function. The second column shows rotated views of the plots in the first column. The third column shows the errors between the original function and its approximation.
  • Figure 5: Approximation of the function $g$, Eq. \ref{['Trig']}, using PU-MLS and DDPU-MLS with the Wendland $\mathcal{C}^2$ function. The second column shows rotated views of the plots in the first column. The third column shows the errors between the original function and its approximation.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 2.1: W
  • Theorem 2.1: W
  • Definition 2.2: MLS pointwise approximation
  • Theorem 2.2: W
  • Definition 2.3: W
  • Lemma 2.3: RRYW
  • Theorem 2.4: W
  • Corollary 2.5: W
  • Definition 3.1: W
  • Definition 3.2: F
  • ...and 3 more