Data-dependent Compact Multinode Shepard Method
Abstract
We introduce a data-dependent compact multinode Shepard method for the approximation of scattered multivariate data. The proposed approach extends the compact multinode Shepard operator by incorporating nonlinear, data-dependent weights inspired by the weighted essentially non-oscillatory (WENO) methodology while preserving the partition of unity structure and the locality of the approximation. The construction is based on compactly supported multinode weight functions associated with local polynomial interpolants defined on unisolvent stencils. A smoothness indicator is employed to adapt the contribution of each stencil according to the local regularity of the underlying function, thereby reducing spurious oscillations near discontinuities while retaining the optimal approximation order in smooth regions. We establish the theoretical approximation properties of the proposed operator: it achieves the order of accuracy of its linear counterpart for sufficiently smooth functions and, under an explicit indicator-separation hypothesis, the same order is retained at every covered evaluation point near a jump discontinuity. Numerical experiments for local degrees up to four illustrate the performance of the method on both smooth and non-smooth data, on scattered as well as on gridded nodes.
Related
- Efficient and Fast Tensor-Product Multinode Shepard Collocation for Elliptic PDEs on Cartesian Grids
- Scattered data interpolation on the torus by compactly supported multinode Shepard operators
- The interpolation method for the complex Hessian equation on compact Kähler manifolds
- Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws
- A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes
BibTeX
@misc{dellaccio2026datadependent,
title = {{Data-dependent Compact Multinode Shepard Method}},
author = {Francesco Dell'Accio and Filomena Di Tommaso and Francesco Larosa and Juan Ruiz-Álvarez and Dionisio F. Yáñez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31026},
eprint = {2610.31026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31026},
note = {CC0},
}
RIS
TY - UNPB
TI - Data-dependent Compact Multinode Shepard Method
AU - Francesco Dell'Accio
AU - Filomena Di Tommaso
AU - Francesco Larosa
AU - Juan Ruiz-Álvarez
AU - Dionisio F. Yáñez
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31026
UR - https://arxiv.org/abs/2610.31026
AB - We introduce a data-dependent compact multinode Shepard method for the approximation of scattered multivariate data. The proposed approach extends the compact multinode Shepard operator by incorporating nonlinear, data-dependent weights inspired by the weighted essentially non-oscillatory (WENO) methodology while preserving the partition of unity structure and the locality of the approximation. The construction is based on compactly supported multinode weight functions associated with local polynomial interpolants defined on unisolvent stencils. A smoothness indicator is employed to adapt the contribution of each stencil according to the local regularity of the underlying function, thereby reducing spurious oscillations near discontinuities while retaining the optimal approximation order in smooth regions. We establish the theoretical approximation properties of the proposed operator: it achieves the order of accuracy of its linear counterpart for sufficiently smooth functions and, under an explicit indicator-separation hypothesis, the same order is retained at every covered evaluation point near a jump discontinuity. Numerical experiments for local degrees up to four illustrate the performance of the method on both smooth and non-smooth data, on scattered as well as on gridded nodes.
ER -