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Data-dependent Compact Multinode Shepard Method

Francesco Dell'Accio, Filomena Di Tommaso, Francesco Larosa, Juan Ruiz-Álvarez, Dionisio F. Yáñez

arXiv:2610.31026v1math.NAcs.NA

TL;DR

Piecewise-smooth data can contaminate local stencils and cause oscillations in scattered-data approximation. This paper adds residual-based nonlinear weights to compact multinode Shepard approximation; it retains the smooth-region order and, under indicator separation, the same order near a jump.

  • Problem

    In linear multinode Shepard approximation, active stencils contribute according to geometric weights regardless of data regularity, although mixed-branch stencils can produce oscillations near jumps.

    Source: §1 · p. 2

  • Method

    The method modifies compact multinode weights using WENO-inspired smoothness indicators computed from residuals at data sites outside each interpolation stencil.

    Source: §1 · p. 2

  • Results

    For the circular discontinuity on Halton nodes (N = 652), the nonlinear operator improves on the linear one by a factor between 45 and 2.3 · 10^3 at distance larger than H from Γ; the same O(h^d+1) order holds near jumps under indicator separation.

    Source: §1 · p. 2, §7 · p. 22

  • Takeaways & Limitations

    Nonlinear renormalization preserves the partition of unity, interpolation property, guaranteed regularity class, and convex-combination structure of the linear operator.

    Source: §7 · p. 21

    The near-jump rate is pointwise conditional, and uniform validity would require a covering-regularity property for the selected stencils that remains an open problem.

    Source: §5.4 · p. 16

Abstract

from arXiv · show

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.

1. Introduction

Higher-degree local stencils can mix smooth branches near jumps, causing poor local representations and oscillations. The paper adds residual-based nonlinear weights to compact multinode Shepard approximation and analyzes when its accuracy is retained.

1. Introduction

  • Stencils mixing data from different smooth branches can produce pronounced oscillations near discontinuities, while linear weights ignore stencil regularity.The paper asks whether data-sensitive weights can reduce nonsmooth contributions without affecting smooth-region approximation.Source: §1 · p. 2
  • The method combines compact multinode Shepard approximation with WENO-inspired nonlinear weights based on each stencil’s smoothness indicator.The indicator uses residuals at data sites outside the interpolation stencil, requiring no additional function evaluations.Source: §1 · p. 2
  • Under uniform locality and stability assumptions, the operator retains O(h^d+1) accuracy for sufficiently smooth functions, regardless of the nonlinear exponent.Near a jump, the same order is proved only under indicator separation; an explicit configuration shows the residual indicator can miss a crossing stencil.Source: §1 · p. 2
  • Experiments compare linear and data-dependent operators on smooth and piecewise smooth data for local polynomial degrees up to four.They also investigate the nonlinear exponent.Source: §1 · p. 2

2. Classical and compact Shepard methods

Classical Shepard approximation uses normalized inverse-distance weights on data values, while compact support localizes their influence. Its partition-of-unity and interpolation properties coexist with constant-only reproduction and parameter-dependent regularity.

2. Classical and compact Shepard methods

  • Classical Shepard approximation is a normalized inverse-distance weighted average of data values and reproduces only constants.Its partition-of-unity structure and continuous extension at nodes make it interpolatory.Source: §2 · p. 3
  • The regularity of the continuously extended Shepard weights depends on μ, ranging from C0 to C∞ for the listed parameter cases.For even integer μ greater than one, the weights are C∞ rational functions with no real poles.Source: §2 · p. 3
  • Compactly supported weights restrict the influence of distant data sites, with support radii typically chosen to contain a prescribed number of sites.Source: §2 · pp. 3–4

3. Multinode Shepard method and compact multinode Shepard method

The multinode construction blends local polynomial interpolants on unisolvent node stencils, and its compact version localizes their weights. A strict fill-distance bound ensures the supports cover the domain.

3. Multinode Shepard method and compact multinode Shepard method

  • The multinode method replaces individual data values with degree-d polynomial interpolants on unisolvent stencils, enabling higher approximation orders on smooth functions.The interpolants are blended using weights with a partition-of-unity structure.Source: §3 · p. 4
  • The compact multinode method replaces global weights with localized weights whose support is the intersection of balls centered at stencil nodes.The support radii are specified using a strict upper bound h on the fill distance.Source: §3 · pp. 4–5
  • At a covered point, the CMS operator blends the local interpolants associated with its active stencils.A point is covered when at least one stencil support contains it.Source: §3 · p. 5
  • Taking h strictly larger than the fill distance ensures the stencil supports cover Ω, so the CMS operator is defined throughout the domain.Equality can leave fill-distance maximizers on support boundaries, where the operator may be undefined.Source: §3 · p. 5

4. Data-dependent compact multinode Shepard method: construction

The construction modifies compact multinode weights using data-dependent smoothness indicators, then renormalizes them. Indicators use residuals beyond each interpolation stencil, and their behavior differs between smooth regions and stencils near jumps.

4. Data-dependent compact multinode Shepard method: construction

  • The nonlinear extension incorporates WENO-inspired data-dependent weights into the compact multinode Shepard method.Source: §4 · p. 6
  • The exponent t controls damping of nonsmooth contributions, while ε keeps the nonlinear-weight denominator positive; the asymptotic analysis takes ε ≤ h^d+1.Source: §4 · p. 6
  • Each stencil’s smoothness indicator uses residuals at nodes outside its interpolation stencil, because residuals there vanish identically.Those nodes are already in the data set, so evaluating the same local interpolant there requires no additional function evaluations.Source: §4 · p. 6
  • In smooth regions the indicator is O(h^d+1); it is bounded away from zero for a one-sided stencil whose indicator set reaches the opposite side of a discontinuity.That lower bound does not extend to arbitrary stencils crossing the discontinuity.Source: §4 · p. 6

5. Properties of the new operator

This section examines the operator’s structural and approximation properties, including polynomial reproduction, continuity, and retention of the linear method’s order in smooth regions and near discontinuities under explicit indicator-separation assumptions. It also describes the nonlinear weights’ reduced contributions from stencils detected as non-smooth and experiments on the circular jump, while noting that the theorem’s conditions are not known to hold uniformly.

5. Properties of the new operator

  • Locality and uniform conditioning of the stencils yield bounded interpolation Lebesgue constants and an O(h^{d+1}) local polynomial interpolation error.The bounds rely on Assumption 5.1 and quasi-uniform nodes; the stencil construction enforces the scaled Vandermonde conditioning threshold.Source: §5 · pp. 7–8

5.1. Detection properties of the indicator

  • The indicator is small for smooth data and separates certain one-sided stencils from a jump when their indicator sets reach the opposite branch within the tubular neighborhood.This detection condition does not cover stencils whose interpolation nodes straddle the jump; in nonconvex domains, the tubular-neighborhood restriction is essential.Source: §5.1 · p. 9

5.2. Structural properties

  • The nonlinear operator uses nonnegative normalized weights, reproduces Πd(Rs) on the covered set, is continuous, and inherits regularity from its compact weights.Away from data sites, this regularity is C⌊µ⌋ for noninteger µ and Cµ−1 for integer µ.Source: §5.2 · p. 10

5.3. Convergence in smooth regions

  • In smooth regions, the operator achieves O(h^{d+1}) accuracy for any ǫ > 0 and t ≥ 1, with no additional restrictions on t or ǫ.All active local interpolants already achieve this order, so any convex combination preserves it.Source: §5.3 · pp. 11–12

5.4. Behaviour near a discontinuity

  • Under Assumptions 5.1 and 5.5, the operator retains the smooth-region order at covered points near a jump for fixed t ≥ 1.The result requires a correct-side reference stencil and separation of every active straddling stencil by the indicator; the indicator alone need not provide that separation.Source: §5.4 · pp. 13–14
  • Any fixed t ≥ 1 preserves the stated rate when 0 < ε ≤ h^(d+1), but the estimates do not determine an optimal finite-resolution choice.Experiments used t = 4, 6, 8, 12 for d = 1, 2, 3, 4, respectively; among tested values, t = 6 minimized error for d = 2, while d = 4 error decreased through t = 12 and was essentially unchanged at t = 14 and 16.Source: §5.4 · p. 16
  • At all four tested levels and every degree, error beyond 2H from Γ was at most 1.2 times the maximum smooth-case error.The band where error exceeded a quarter of the jump had width between 0.6H and 0.8H.Source: §5.4 · p. 16
  • For d ≥ 1, O(h^(d+1)) accuracy holds in smooth regions and at points satisfying Assumption 5.5, but the result is pointwise, not uniform.Establishing coverage regularity for the implemented minimal-covering construction remains open.Source: §5.4 · p. 16
  • Both operators form convex combinations of active local interpolants, but nonlinear weights reduce contributions from stencils detected as non-smooth.Source: §5.4 · p. 16

6. Numerical experiments

Experiments compare linear and data-dependent compact multinode Shepard operators on smooth and piecewise-smooth data, using Halton nodes and regular grids. They test convergence, behavior near jumps across geometries, and sensitivity to the nonlinear exponent.

6.1. Order of accuracy in smooth regions

  • On smooth data, linear and nonlinear operators exhibit the same convergence orders at every degree, with nonlinear RMS rates consistent with order d + 1.For degrees 1–4, finest-step rates are 2.02, 3.12, 3.99, and 5.20; least-squares slopes over four levels are 2.05, 3.09, 3.95, and 4.95.Source: §6.1 · p. 17, Table 1 · p. 18
  • At equal node counts, regular-grid maximum errors are smaller than Halton errors for degrees 1 and 4 at every level, but not consistently for degrees 2 and 3.The nonlinear grid error is larger in the specified degree-2 levels and at every tested level for degree 3.Source: §6.1 · p. 18, Table 2 · p. 19

6.2. Approximation of piecewise smooth functions

  • For the circular jump, nonlinear errors beyond 2H are at most 1.2 times the smooth-case maximum, while linear errors remain between 0.25 and 1.2 beyond H.The nonlinear contaminated band is 0.7–0.8H, versus 1.3H, 1.8H, 2.0H, and 2.7H for the linear operator at degrees 1–4.Source: §6.2 · p. 18, Table 3 · p. 19
  • The nonlinear operator’s contaminated band stays between 0.6H and 0.8H across tested levels and degrees, while the linear operator’s band spans 1.2H to 2.9H.The nonlinear band therefore shrinks proportionally to H over the tested levels.Source: §6.2 · p. 18, Table 4 · p. 20
  • For line and square jumps, the nonlinear operator recovers smooth-case accuracy beyond 2H for every degree, though errors above a quarter of the jump occur at a few points between H and 1.5H.The square’s corner is outside the paper’s C1 setting and serves as a stress test.Source: §6.2 · p. 19
  • On regular grids, degree 4 has a wider contaminated band than on scattered nodes; degrees up to 3 behave comparably across the two node distributions.At fixed physical distances from the jump, degree-4 errors of order 10^-2 persist to 2δH for all three geometries, reaching smooth levels only at 3δH.Source: §6.2 · pp. 20–21

6.3. Sensitivity to the exponent t

  • Among tested exponents, t = 6 is best for degree 2; for degree 4, error decreases through t = 12 and remains at the same level at t = 14 and 16.Source: §6.3 · p. 21

6.4. Reproducibility

  • Experiments use MATLAB and an independently verified Python implementation, with scripts for every table and figure available from the authors.Source: §6.4 · p. 21

7. Conclusions

The data-dependent compact multinode Shepard method combines nonlinear smoothness-based weighting with the compact multinode construction. Experiments show improved accuracy near discontinuities in several tested settings, while the near-jump convergence result is conditional and the observed performance depends on stencil coverage.

7. Conclusions

  • The method preserves locality, interpolation, partition of unity, and convex-combination structure while using local-interpolant residuals on outside-stencil nodes as smoothness indicators, without additional function evaluations.It attains order h^d+1 in smooth regions for any parameter choice.Source: §7 · p. 21
  • Figure 3 separates 101 high-indicator stencils from the others: no indicator falls between 1.35 × 10^-2 and 0.17, leaving the two modes about three orders of magnitude apart.The high-indicator group includes stencils straddling the discontinuity and one-sided stencils detected near it.Source: Figure 3 · p. 22
  • For all tested Halton geometries and degrees, the nonlinear operator recovers smooth-case accuracy beyond 2H; for the circular jump, errors beyond H improve by factors of 45 to 2.3 · 10^3 over the linear operator.On regular grids, the contaminated band is wider for degree 4.Source: §7 · p. 22, Table 6 · p. 23
  • Near a jump, the same O(h^d+1) order holds under an explicit indicator-separation hypothesis, but constants grow exponentially with stencil cardinality and may require larger exponents at higher degrees.The tested line and square geometries show irregular gains by degree, including a case where the nonlinear operator is less accurate.Source: §7 · p. 22
  • Some losses beyond H arise when no one-sided stencil is active or only an opposite-branch one-sided stencil is active, identifying coverage rather than the indicator as the constraint.The authors point to discontinuity-adapted coverings that exclude active straddling stencils.Source: §7 · pp. 22–23
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