Data dependent Shepard approximation through and adaptive modification of the shape parameter

Abstract: In this article, we introduce a novel data-dependent Shepard interpolation method inspired by the adaptive strategies proposed in [2]. In this case, as Shepard interpolation does not produce oscillations, our approach has the core objective of reducing the smearing near jump discontinuities in the data in one and two dimensions. While the original work in [2] focuses in on Radial Basis Function (RBF) interpolation, we extend these ideas to the Shepard framework by incorporating a data-dependent adaptation mechanism. Specifically, we modify the classical Shepard interpolation by adaptively adjusting the influence weights based on local smoothness indicators that modify the shape parameter. These indicators, similar to those used in [2], are designed to detect discontinuities: for grid-based data, we use squared undivided second-order differences, and for scattered data, we employ squared least-squares approximations of the Laplacian scaled by the square of the mean local separation of stencil points. The resulting data-dependent weighting scheme forces the kernels close to a discontinuity to behave like a local delta function, effectively reducing the smearing of the discontinuities introduced by the classical Shepard approach. We establish the theoretical foundation of the method, including the properties of the new interpolation and we theoretically prove that the reduction of the smearing of discontinuities is possible. Numerical experiments in one and two dimensions confirm that the proposed data-dependent Shepard interpolation significantly reduces the smearing of jump discontinuities while maintaining high accuracy in smooth regions.

• The paper introduces an adaptive modification to classical Shepard interpolation that uses local smoothness indicators to adjust RBF shape parameters near discontinuities.
• It reduces smearing belts by approximately a factor of 1/κ, ensuring interpolation accuracy in smooth regions and improved fidelity across jumps.
• Numerical experiments confirm robust performance across multiple kernels and sampling schemes, demonstrating scalability to multidimensional data.

Data-Dependent Shepard Approximation with Adaptive Shape Parameter Modification

Introduction

The paper "Data dependent Shepard approximation through and adaptive modification of the shape parameter" (2606.20332) introduces a novel extension to classic Shepard interpolation focused on reducing smearing near jump discontinuities in one and two dimensions. Shepard interpolation is widely utilized for multivariate function approximation due to its robustness and applicability to scattered data. However, it suffers from smearing artifacts at discontinuities, which degrade interpolation quality for piecewise-smooth functions. The authors leverage advances from adaptive RBF interpolation, specifically the adaptive shape parameter modification, to extend Shepard interpolation with a data-dependent weighting scheme governed by local smoothness indicators.

Methodology

Adaptive Shepard Interpolation

The classical Shepard formula normalizes weighted averages, with weights generally constructed from RBF kernels $\phi(\varepsilon |\mathbf{x}-\mathbf{x}_i|)$ using a fixed shape parameter $\varepsilon$. The proposed method replaces the fixed parameter with a locally adaptive version $\tilde\varepsilon_i$, which increases in the vicinity of detected discontinuities. The resulting interpolant is:

$$\tilde{s}(\mathbf{x}) = \frac{ \sum_{i=1}^N \phi(\tilde{\varepsilon}_i \|\mathbf{x} - \mathbf{x}_i\|) y_i }{ \sum_{i=1}^N \phi(\tilde{\varepsilon}_i \|\mathbf{x} - \mathbf{x}_i\|) }$$

where

$$\tilde{\varepsilon}_i = \varepsilon \cdot \frac{1}{c + g(I_i)}$$

with $c$ a small constant (preventing division by zero), and $g$ a decaying function of the smoothness indicator $I_i$. This formulation ensures that $\tilde{\varepsilon}_i$ becomes large near discontinuities, localizing the RBF kernel and approximating a delta function, while retaining standard interpolation properties in smooth regions.

Smoothness Indicators and Adaptivity

The smoothness indicator $I_i$ is designed to distinguish between smooth and non-smooth regions. For grid-based data, squared undivided second-order differences are used, while for scattered data, squared least-squares Laplacians scaled by the mean local separation are employed. In smooth regions, $\varepsilon$0 scales as $\varepsilon$1; near discontinuities, $\varepsilon$2. The adaptivity mechanism sharply increases $\varepsilon$3 at detected jumps, substantially reducing smearing by enforcing locality.

Theoretical Analysis

The authors rigorously demonstrate that their adaptive strategy can reduce the smearing belt around discontinuities inherent in the classical Shepard approach. Error identities and bounds quantify the influence of kernels from the wrong side of the discontinuity, revealing that the adaptive approach shrinks the smearing belt width by approximately a factor $\varepsilon$4, where $\varepsilon$5 controls localization. Explicit bounds and ratios are provided for both constant and adaptive shape parameter settings, covering cases with exponential decay kernels and compact support.

Numerical Results

Smooth Function Reconstruction

Experiments on analytic functions confirm that the adaptive modification does not degrade interpolation accuracy in smooth regions. Both classical and adaptive Shepard methods produce virtually identical errors, with choices of various RBF kernels yielding stable and reliable results.

Piecewise-Smooth Data with Discontinuities

For piecewise-smooth target functions with jump discontinuities, the adaptive approach achieves strong reduction in smearing near the discontinuities compared to classical Shepard interpolation. The smearing belt contraction is evident for both gridded and Halton-distributed nodes, and robust across all tested kernels (Gaussian, Matérn, Wendland). Numerical error plots validate the theoretical expectations: the adaptive kernels sharply localize around discontinuity points, resulting in minimal artifact spread.

Bivariate Data and Surface Approximation

In two-dimensional tests, including discontinuous modifications of Franke's function, adaptive Shepard interpolation diminishes error spread along jump curves for both regular and scattered datasets. The methodology is directly extensible to multidimensional settings by employing least-squares Laplacian-based smoothness indicators for mesh-free data. Despite some residual smoothing at the discontinuity, the adaptive method outperforms the classical scheme in maintaining interpolation fidelity both locally and globally.

Implications and Extensions

The data-dependent modification of Shepard interpolation advances the robust approximation of piecewise-smooth functions, particularly where accurate recovery near discontinuities is essential. This is significant for applications in science and engineering involving physical interfaces, image reconstruction, and data analysis on irregular domains. The method is agnostic to sampling regularity and scalable to higher dimensions.

From a theoretical perspective, the framework generalizes to other kernel-based interpolation schemes, suggesting broader applicability to mesh-free PDE solvers, non-uniform data processing, and adaptive approximation algorithms. Integration with automated discontinuity detection or hybrid smoothness indicators could enhance localization further. Future research may focus on optimizing the functional form of $\varepsilon$6, exploring computational acceleration for large-scale problems, and analyzing stochastic or adversarial data distributions.

Conclusion

The presented data-dependent Shepard interpolation with an adaptive shape parameter (2606.20332) addresses persistent smearing issues in classical interpolation of discontinuous and piecewise-smooth data. By automatically localizing weights near jumps, the method minimizes cross-side influence, narrowing the smearing belt and preserving accuracy in smooth regions. Numerical results and theoretical proofs corroborate the effectiveness of this approach across diverse kernels and sampling schemes. This methodology offers a principled, flexible framework for stable recovery of functions with discontinuities and is compatible with broader mesh-free and kernel-based approximation contexts.