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Scattered data interpolation on the torus by compactly supported multinode Shepard operators
F. Dell'Accio, F. Di Tommaso, R. Lammirato, F. Larosa
TL;DR
The paper addresses scattered-data interpolation on a torus while avoiding redundant polynomial representations and global stencil contributions. It constructs a compactly supported multinode Shepard operator with Gröbner-reduced local spaces and justified Euclidean supports, obtaining order d+1 convergence under uniform locality and stability assumptions. Numerical tests confirm reproduction and error reduction, and CFD experiments reconstruct velocity quantities and tangent fields.
Problem
Polynomial restrictions on the torus are redundant because ambient polynomials differing by multiples of the quartic defining equation represent the same surface function, while local toroidal distances lack an equally convenient geodesic formula.
Method
The method combines Gröbner-reduced toroidal polynomial spaces, minimal local interpolation stencils, compactly supported multinode Shepard weights, and locally justified ambient Euclidean distances.
Results
Under uniform locality and stability assumptions, the method converges with order d+1 in the fill distance; experiments confirm reproduction, error reduction, and CFD reconstruction of velocity components, magnitude, and tangent fields.
Takeaways & Limitations
Compactly supported multinode Shepard operators provide an effective interpolatory meshfree tool for scattered data on toroidal geometries.
Takeaways & Limitations
The convergence result assumes uniform locality and bounded local Lebesgue factors, and fixed-degree h-convergence does not establish degree-enrichment behavior at a fixed node set.
Abstract
from arXiv · showhide
We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.
1 Introduction
The paper extends multinode Shepard interpolation from the sphere to the torus by combining reduced toroidal polynomial spaces, minimal local stencils, and compactly supported weights. Gröbner reduction handles algebraic redundancy, while local metric equivalence supports genuinely local construction and convergence analysis.
- Scattered interpolation reconstructs functions from values at irregular points, motivating meshfree methods for complex and non-Euclidean geometries.
- Multinode Shepard operators improve on classical Shepard approximation by using local polynomial interpolants instead of only normalized weighted averages.
- The torus extension retains minimal unisolvent local interpolants but replaces spherical-harmonic representation with Gröbner-reduced polynomial bases for the quartic surface equation.
- The periodic parameter distance and embedded Euclidean distance are locally equivalent, justifying sufficiently small Euclidean supports on the torus.
- Compact support truncates inverse-distance factors so only locally active stencils contribute, while support radii preserve coverage, interpolation, and polynomial reproduction.
- The contributions include Gröbner-based restriction spaces, Leja-type minimal stencils, compact support with coverage guarantees, order d+1 convergence, and analytical and CFD experiments.
2 Gröbner-based polynomial spaces on the torus
The torus interpolation space is formed by restricting ambient polynomials and removing redundancies caused by the quartic defining equation. Gröbner normal forms produce an explicit basis whose size determines the local Vandermonde systems and interpolation stencils.
- The quartic torus equation creates dependent restrictions among degree-d trivariate polynomials, making unreduced bases potentially singular for interpolation.
- The torus is represented both by a periodic parametrization in R2/(2πZ)2 and as a real algebraic surface in R3.
- The relevant space Hd(T) consists of restrictions to the torus of trivariate polynomials of total degree at most d.
- Gröbner reduction modulo the principal ideal generated by F gives unique normal forms using monomials not divisible by x4.
- Restrictions of the reduced monomials form a basis of Hd(T), with dimension md; the experiments use m1=4, m2=10, m3=20, m4=34, m5=52, and m6=74.
- A node set of cardinality md is unisolvent when its Vandermonde matrix is nonsingular, yielding a unique interpolant for every data vector.
3 The multinode Shepard framework
The torus construction blends local polynomial interpolants on unisolvent stencils with distance-based multinode Shepard weights. Local equivalence of periodic and ambient distances justifies compact supports, while coverage and stability yield interpolation, polynomial reproduction, and convergence guarantees.
- Multinode Shepard construction: Multinode Shepard weights blend the local interpolants through a partition of unity while enforcing interpolation at the nodes.The operator is defined piecewise at interpolation nodes and satisfies the partition-of-unity property away from them.
- Multinode Shepard construction: Local interpolants Pj[f] are constructed on stencils unisolvent for the restriction space Hd(T).Each stencil has a unique interpolant in Hd(T), and the global operator combines these local approximations.
- Local comparison of distances: The periodic parameter distance and embedded Euclidean distance are locally equivalent, although they can differ substantially for globally separated parameter points.The local result supports ambient-distance weighting, while the half-turn example exhibits an embedding-induced shortcut.
- Compact support: Compactly supported weights restrict each evaluation to nearby active stencils, with radii based on stencil diameters and an upper fill-distance bound.Choosing H greater than the fill distance ensures every evaluation point has an active stencil, keeping the normalized denominator positive.
- Error estimate and convergence: The operator reproduces Hd(T), and for fixed degree d its error estimate applies under sufficiently small support radii, uniform stencil locality, and bounded local Lebesgue constants.The convergence result does not require a lower threshold on the weight exponent µ; nonsingularity alone does not guarantee uniform stability.
4 Numerical experiments on analytical data
The experiments evaluate the compactly supported multinode Shepard operator on analytical torus functions under node refinement, degree enrichment, and high-degree stability diagnostics. Results confirm polynomial reproduction, decreasing interpolation errors, and near-machine-precision local computations despite ill-conditioned monomial systems.
- Experimental setup: The study uses 15 analytical torus functions, including polynomial and smooth non-polynomial examples, to assess reproduction and approximation behavior.The functions are visualized through radial surface deformations and color maps before interpolation tests.
- Experimental setup: Halton nodes with bases 2 and 3 are generated in the periodic parameter domain and mapped onto the torus.The mapped nodes are low-discrepancy in parameter space but are not generally equidistributed under torus surface area.
- Node refinement: With d = 4 fixed, increasing the node count from N = 1000 to N = 64000 systematically reduces maximum, mean, and RMS interpolation errors.A direct experimental order would additionally require fill distances and experimental convergence rates.
- Degree enrichment: At fixed N = 64000, increasing d from 1 to d = 6 generally reduces errors, with improvement for non-polynomial functions depending on regularity and complexity.Polynomial targets reach machine precision once the local space contains the target function.
- High-degree diagnostics: For d = 5 and d = 6, local solves retain near-machine-precision backward errors and moderate perturbation amplification, although monomial coefficient systems become increasingly ill-conditioned.The diagnostics do not establish a uniform high-degree stability bound; sampled Lebesgue factors are more relevant for operator-level sensitivity.
5 Application to Computational Fluid Dynamics data on the torus
The method is applied to CFD velocity data mapped from a periodic square grid onto a torus, reconstructing scalar components, velocity magnitude, and the tangent field. Using 1800 training nodes and 200 validation nodes, the interpolations achieve small reported errors.
- Data mapping: The CFD dataset contains a two-dimensional velocity field on a periodic square domain, mapped to the torus using α = 2πx and β = 2πy.The torus parameters are R = 3 and r = 1, and the velocity components are interpreted in the orthonormal tangent frame.
- Experimental setup: The experiments use 2000 data points, with 1800 training nodes, 200 validation nodes, and 244 generated local interpolation stencils.Training nodes construct the interpolants and weights, while validation nodes are reserved for accuracy assessment.
- Scalar reconstruction: The component u has mean error 3.4561e-6 and maximum error 5.5247e-5, while v has mean error 1.3859e-6 and maximum error 1.6019e-5.The velocity magnitude is reconstructed from the separately interpolated components rather than interpolated independently.
- Magnitude reconstruction: The reconstructed velocity magnitude has maximum, mean, and RMS errors bounded by the corresponding vector-field errors, consistent with the numerical values in Table 4.This follows from the reverse triangle inequality applied at validation nodes.
- Tangent-field reconstruction: The tangent-field reconstruction reports Emean = 4.0557e-6, Emax = 5.5268e-5, and ERMS = 8.1895e-6.The two scalar interpolants are combined through the orthonormal tangent frame; the plots indicate preservation of direction and magnitude.
6 Implementation and LU-based stencil construction
The implementation constructs reduced toroidal Vandermonde systems, extracts minimal stencils by LU row pivoting, and evaluates compactly supported blends using exact geometric activation tests and stable logarithmic normalization.
- Implementation overview: The implementation has three stages: reduced toroidal basis generation, covering-family stencil selection, and stable evaluation of the compactly supported blend.The implementation extends a MATLAB framework for multinode Shepard interpolation.
- Local systems: The reduced toroidal Vandermonde matrix evaluates translated and scaled reduced-basis functions on candidate points, yielding a square system when the selected functions are independent.The translated and scaled functions remain restrictions of ambient polynomials of total degree at most d.
- Basis construction: The basis-generation algorithm lists monomials of total degree at most d and removes those divisible by LM(F) = x^4 before ordering the reduced basis.This algebraic reduction removes redundancy induced by the torus defining polynomial.
- Stencil construction: Partial row pivoting in the rectangular candidate matrix selects the first md pivot rows as a minimal stencil, while selected indices are removed from the uncovered list.Candidate sets are drawn from the full node set, so different stencils may overlap.
- Candidate selection: Candidate neighborhoods are enlarged by ten-percent increments until they contain at least md+1 = 2((d + 1)^2 + 1) nodes, which are then ordered by ambient Euclidean distance.The initial neighborhood is an axis-aligned box in R3.
- Stable evaluation: A stencil is active only when dE(x, xjk) < Rj for every stencil node, with Rj = diamE(σj)+H; normalized weights are evaluated using logarithmic stabilization.At data nodes the prescribed value is returned directly, and the final blend uses log-sum-exp normalization.
7 Conclusion
The paper presents a compactly supported multinode Shepard method tailored to toroidal geometry through algebraic basis reduction and local Euclidean supports. It proves order d + 1 convergence under locality and stability assumptions and reports accurate analytical and CFD reconstructions, while noting that numerical evidence does not prove uniform high-degree stability.
- Conclusion: The method combines minimal local polynomial interpolation, compactly supported partition-of-unity weights, and a Gröbner-reduced basis adapted to the torus.Euclidean distances are justified within sufficiently small supports by local equivalence with periodic parameter distance.
- Theory: Under uniform locality and stability assumptions, the method converges with order d + 1 with respect to the fill distance.The estimate is established using a smooth normal extension and a local ambient Taylor argument.
- Numerical evidence: The experiments confirm polynomial reproduction and reduced interpolation errors under node refinement and degree enrichment.Reduced-space modes are reproduced nearly to machine precision, and tested perturbation amplification remains moderate.
- Scope boundary: The diagnostics provide evidence of high numerical accuracy for smooth data but do not prove uniform high-degree stability.Monomial representations become increasingly ill-conditioned at higher degree even though factorization and solve backward errors remain near machine precision.
- CFD application: The CFD application accurately reconstructs velocity components, velocity magnitude from interpolated components, and tangent velocity fields through orthonormal lifting.These results support the method as an interpolatory meshfree tool for scattered data on toroidal geometries.
A Generation of the reduced toroidal basis
The reduced toroidal basis is generated by enumerating ambient monomials up to total degree d, excluding monomials divisible by the leading monomial x^4 of the torus equation, and printing the remaining monomials.
- Monomial enumeration: The procedure loops over exponents satisfying i + j + k ≤ d to enumerate all candidate monomials.The exponent loops generate the x, y, and z factors for each monomial.
- Monomial construction: It appends x, y, and z factors with the appropriate exponent notation to construct each monomial string.Exponent-one factors are printed without a superscript, while higher powers use ^ notation.
- Output: When no factors are present, the procedure represents the constant monomial as 1; otherwise it concatenates the collected factors.Each constructed monomial is then printed and counted.
Declarations
The authors report no conflict of interest, identify project funding and institutional affiliations, describe equal collaborative contributions, and state data-availability provisions.
- The authors declare that they have no conflict of interest.
- The research was supported by the GNCS-INdAM 2026 project.The project concerns polynomial and kernel methods for approximation from discrete and integral data with OS software.
- All four authors contributed equally to the conception and scientific development of the work.The listed activities were carried out collaboratively, including analysis, implementation, experimentation, validation, interpretation, and manuscript preparation and revision.
- The research was carried out within RITA, the UMI approximation group, and the INdAM-GNCS Research Group.
- The CFD data are cited in the manuscript, while implementation files and derived data are available upon reasonable request.