Weighted Essentially Non-Oscillatory Shepard method

TL;DR

This work addresses interpolation of scattered data in multiple dimensions in the presence of piecewise-smooth functions with discontinuities. It introduces a nonlinear WENO-inspired Shepard method (WENO-S) that replaces the linear Shepard weights with nonlinear weights $\mathcal{W}_i(\mathbf{x}) = \alpha_i(\mathbf{x}) / \sum_j \alpha_j(\mathbf{x})$ where $\alpha_i(\mathbf{x}) = W_i(\mathbf{x}) / (\epsilon + I_i)^t$ and $I_i$ are smoothness indicators computed from local least-squares fits; the interpolant is $\mathcal{I}_{\text{WENO-S}}(\mathbf{x}) = \sum_i \mathcal{W}_i(\mathbf{x}) f_i$. The authors prove that the method preserves smoothness properties (it lies in $C^\nu$ if the kernel $\omega$ is in $C^\nu$) and provide an $O(h)$ error bound in smooth regions, while showing a significant reduction in diffusion near discontinuities. Numerical experiments with Franke's function and discontinuity curves demonstrate comparable performance to linear Shepard in smooth zones and improved sharpness near interfaces, highlighting the method's robustness for scattered-data interpolation in challenging, piecewise-smooth scenarios.

Abstract

Shepard method is a fast algorithm that has been classically used to interpolate scattered data in several dimensions. This is an important and well-known technique in numerical analysis founded in the main idea that data that is far away from the approximation point should contribute less to the resulting approximation. Approximating piecewise smooth functions in $\mathbb{R}^n$ near discontinuities along a hypersurface in $\mathbb{R}^{n-1}$ is challenging for the Shepard method or any other linear technique for sparse data due to the inherent difficulty in accurately capturing sharp transitions and avoiding oscillations. This letter is devoted to constructing a non-linear Shepard method using the basic ideas that arise from the weighted essentially non-oscillatory interpolation method (WENO). The proposed method aims to enhance the accuracy and stability of the traditional Shepard method by incorporating WENO's adaptive and nonlinear weighting mechanism. To address this challenge, we will nonlinearly modify the weight function in a general Shepard method, considering any weight function, rather than relying solely on the inverse of the distance squared. This approach effectively reduces oscillations near discontinuities and improves the overall interpolation quality. Numerical experiments demonstrate the superior performance of the new method in handling complex datasets, making it a valuable tool for various applications in scientific computing and data analysis.

Figures (3)

• Figure 1: Approximation to function $\tilde{f}$, Eq. \ref{['funciontildef']}. In first line, a 3-D plot with a circumference as discontinuity curve and the next two rows in a 2-D plot with different discontinuity curves using regular grid data points.
• Figure 2: Approximation to function $\tilde{f}$, Eq. \ref{['funciontildef']}, in a 2-D plot using regular grid data points.
• Figure 3: Approximation to function $\tilde{f}$, Eq. \ref{['funciontildef']}, in a 2-d plot with different discontinuity curves using Halton scattered data points.

Theorems & Definitions (5)

• Remark 2.1
• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Theorem 3.3