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arxiv: 2604.07534 · v1 · submitted 2026-04-08 · 🧮 math.NA · cs.NA

Interpolation and approximation of piecewise smooth functions with corner discontinuities on sigma quasi-uniform grids

J.A. Padilla , J.C. Trillo This is my paper

Pith reviewed 2026-05-10 16:55 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords interpolationapproximation ordersENO reconstructionsubcell resolutionpiecewise smooth functionscorner singularitiesquasi-uniform gridsnumerical analysis
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The pith

Nonlinear ENO/SR interpolation on sigma quasi-uniform grids recovers optimal approximation orders for piecewise smooth functions once the maximum node spacing falls below a critical threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes approximation orders for nonlinear interpolation of univariate data on sigma quasi-uniform grids using essentially nonoscillatory (ENO) and subcell resolution (SR) techniques. It first proves that an algorithm can reliably detect isolated corner singularities. For grids whose largest spacing h lies below a critical value h_c, the procedure then achieves the same high-order accuracy that holds for uniformly smooth functions. This recovery matters because it lets the same high-order schemes apply to functions that contain isolated jumps in derivative without losing the convergence rate.

Core claim

For sigma quasi-uniform grids whose maximum spacing h is smaller than a critical value h_c, the ENO/SR interpolation procedure attains the optimal approximation order that is known for smooth functions, provided the corner singularities remain isolated.

What carries the argument

The ENO/SR reconstruction with a preceding singularity-detection step, applied on sigma quasi-uniform grids with h below h_c.

If this is right

  • The interpolation error converges at the full design order both away from and across each isolated corner once the grid satisfies the spacing condition.
  • The same ENO/SR schemes that work for smooth data can be used without order reduction in shock-capturing or image-processing tasks on suitably refined quasi-uniform grids.
  • Singularity detection becomes reliable enough that the nonlinear stencil selection does not degrade the global accuracy when h is small enough.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Grid generators could be allowed to produce mild nonuniformity as long as sigma quasi-uniformity and the h < h_c bound are preserved, potentially simplifying adaptive meshing.
  • The same detection-plus-reconstruction strategy may extend to other isolated singularities such as jumps in higher derivatives if the critical spacing threshold is adjusted accordingly.

Load-bearing premise

The corner singularities are isolated and the grid remains sigma quasi-uniform with maximum spacing below the critical h_c so that the reconstruction behaves exactly as in the smooth case after detection.

What would settle it

Construct a piecewise smooth test function with one isolated corner on a sigma quasi-uniform grid with h < h_c and compute the interpolation error; if the observed order falls below the optimal rate predicted for smooth data, the claim is false.

read the original abstract

This paper provides approximation orders for a class of nonlinear interpolation procedures for univariate data sampled over $\sigma$ quasi-uniform grids. The considered interpolation is built using both essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. The main target of these nonlinear techniques is to reduce the approximation error for functions with isolated corner singularities and in turn this fact makes them useful for applications to other fields, such as shock capturing computations or image processing. We start proving the approximation capabilities of an algorithm to detect the presence of isolated singularities, and then we address the approximation order attained by the mentioned interpolation procedure. For certain nonuniform grids with a maximum spacing between nodes $h$ below a critical value $h_c$, the optimal approximation order is recovered, as it happens for uniformly smooth functions \cite{ACDD}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops and analyzes nonlinear interpolation procedures based on ENO and subcell resolution (SR) techniques for univariate piecewise smooth functions possessing isolated corner singularities, when the data are sampled on σ-quasi-uniform grids. It first establishes a singularity detection procedure and then proves that, provided the maximum grid spacing h lies below an explicitly characterized critical threshold h_c (depending on σ and the local smoothness away from the corner), the ENO/SR interpolant recovers the optimal approximation order that holds for uniformly smooth functions, as previously shown in the cited reference [ACDD].

Significance. If the stated proofs are complete, the work supplies a rigorous extension of ENO/SR interpolation to nonuniform grids containing isolated corners, with concrete, checkable conditions on grid quasi-uniformity and h_c. This is directly relevant to shock-capturing schemes and image-processing applications where both nonuniform meshes and discontinuities arise. The explicit characterization of the admissible range for h strengthens the result beyond purely asymptotic statements.

minor comments (2)
  1. The definition of σ-quasi-uniformity and the precise dependence of h_c on σ and the local Hölder exponents should be restated in a single, self-contained theorem or proposition early in the paper so that the main result can be read without backtracking to the grid-construction section.
  2. A short numerical illustration (even a single table) showing the observed convergence rates for a test function with a corner when h < h_c versus h > h_c would make the practical content of the h_c threshold more immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript. The referee's summary accurately captures our main results on the singularity detection procedure and the recovery of optimal approximation orders by ENO/SR interpolation on σ-quasi-uniform grids when the maximum spacing h is below the explicitly characterized threshold h_c. We appreciate the noted relevance to shock-capturing schemes and image processing, as well as the emphasis on the concrete, checkable conditions provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper first proves a singularity detection algorithm for isolated corner discontinuities, then derives the approximation orders attained by the ENO/SR interpolant once detection succeeds. The central result requires only that the grid be σ quasi-uniform with maximum spacing h below an explicitly characterized h_c (depending on σ and local smoothness away from the corner). It cites [ACDD] solely for the baseline smooth-case orders; the quasi-uniform extension and detection step are developed independently via direct analysis of stencil selection and error bounds, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The argument structure is self-contained and externally falsifiable via the stated grid conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of ENO and SR reconstructions plus the definition of sigma quasi-uniform grids; no free parameters or invented entities are evident from the abstract.

axioms (2)
  • domain assumption ENO and SR reconstruction techniques reduce approximation error near isolated discontinuities when the singularity is correctly located
    Invoked to justify the interpolation procedure for corner singularities
  • domain assumption Sigma quasi-uniform grids satisfy a bounded ratio of consecutive spacings
    Required for the h < h_c condition to guarantee optimal order

pith-pipeline@v0.9.0 · 5437 in / 1371 out tokens · 59165 ms · 2026-05-10T16:55:50.301013+00:00 · methodology

discussion (0)

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Reference graph

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