A scale-covariant pre-solve screening algorithm for scalar finite-difference discretizations on positive nonuniform meshes

Technical Summary: A Scale-Covariant Pre-Solve Screening Algorithm for Scalar Finite-Difference Discretizations

Problem Statement
Selecting a nonuniform finite-difference discretization (mesh and stencil) for scalar problems on positive domains ($0 < x_0 < \dots < x_N$) typically occurs before a reference solution is available. Existing decision rules often rely on local consistency metrics (e.g., step sizes, Péclet numbers) or additive Fourier analysis, which are tied to translation invariance. However, these metrics do not directly measure how assembled nonuniform rows transmit multiplicative components (power-law or log-periodic oscillations) declared by the problem. The challenge is to develop a decision rule that uses assembled rows to reject unsafe discretizations before solving, distinguishing between local consistency and global stability, without confusing additive and multiplicative geometries.

Methodology
The paper proposes a scale-covariant pre-solve screening algorithm that evaluates assembled rows against declared multiplicative characters $x^z$. The workflow is divided into a deterministic audit and a risk-calibrated decision branch.

1. Mellin Audit (Deterministic Component):

   • Local Response: For a declared exponent $z$, the algorithm computes the local Mellin response of row $j$:
     $$m_{h,j}(z) = x_j^{-z} \sum_{k \in S_j} a_{jk} x_k^z = \sum_{k \in S_j} a_{jk} \exp\left(z \log \frac{x_k}{x_j}\right)$$
     where $a_{jk}$ are assembled weights and $S_j$ is the row support.
   • Defect Calculation: It compares this discrete response to the continuous operator response $m_j(z)$, forming a defect $d_j(z) = m_{h,j}(z) - m_j(z)$.
   • Aggregation: The audit aggregates normalized defects, inverse responses, phase errors, and spatial roughness over a declared working exponent set $\Pi$. These form a diagnostic vector $\mathbf{M}$.
   • Theoretical Bounds: The method establishes that for homogeneous operators, the normalized response is dilation covariant. It provides a bound on the difference between this multiplicative probe and a standard additive Fourier probe, showing they coincide to first order in relative mesh width but diverge for large steps.

2. Certificate-First Decision Rule:
   The algorithm prioritizes rigorous mathematical certificates over empirical heuristics:

   • Hard Certificate: If the discretized matrix is a Z-matrix and a constructive positive barrier vector $w_h$ (verifiable without solving) exists such that $A_h w_h \ge 1$, the method proves the matrix is a nonsingular M-matrix. Combined with a finite Mellin representation of the solution, this yields a computable, rigorous maximum-norm error bound.
   • Empirical Branch: If a hard certificate cannot be verified, the algorithm falls back to a risk-calibrated reject/defer rule. This branch uses machine learning models (logistic regression, gradient boosting, random forests) trained on features combining conventional mesh indicators ($\mathbf{B}$), additive audit features ($\mathbf{A}$), and Mellin audit features ($\mathbf{M}$).
   • Calibration: The rejection threshold is determined via split-conformal calibration on disjoint groups of physical parameters, ensuring a pre-specified upper bound on the probability of wrongly rejecting a safe candidate (False Positive Rate) under the assumption of score exchangeability.

Key Contributions

• Scale-Covariant Diagnostic: Introduction of a pre-solve audit based on the Mellin–Fourier correspondence, specifically designed for positive nonuniform meshes where dilation acts as translation in log-coordinates.
• Rigorous Separation of Branches: A workflow that strictly separates deterministic, certified acceptance (via M-matrix theory) from empirical, risk-calibrated rejection. The empirical branch is explicitly labeled as "defer" or "reject" rather than "safe" unless a hard certificate exists.
• Theoretical Guarantees:
  • Proof of dilation covariance for homogeneous operators.
  • A finite-net certification theorem (Theorem 6) that bounds the defect over a continuous exponent band using derivative estimates.
  • An exact modal error identity for uniform logarithmic tensor grids (Theorem 7).
  • A constructive barrier certificate for nonsingular M-matrices (Theorem 8).
• Incremental Information Analysis: A rigorous evaluation showing that Mellin features provide statistically significant incremental information for logistic and gradient-boosting models, particularly for scale-structured problems (Euler family), though they are often redundant with strong additive controls for layer-dominated problems.

Results
The method was validated on 1,764 one-dimensional discretizations and 140 tensor-product cases across Euler, reaction-diffusion, and convection-diffusion families.

• Certification: The hard certificate successfully accepted 55–60% of truly safe cases in M-matrix scenarios without solving, with zero false acceptances of unsafe cases.
• Empirical Screening:
  • Adding Mellin features to a gradient-boosting baseline increased the unsafe rejection rate (True Positive Rate) from 76.9% to 80.5% while maintaining a wrong rejection rate (False Positive Rate) of 1.6% (calibrated to a 3% budget).
  • This performance nearly matched a post-solve two-grid difference estimator (80.7% unsafe rejection) but required no refined solution.
  • The Mellin audit showed specific structural advantages for power-law and log-periodic components, where it is exactly dilation covariant.
• External Stress Tests:
  • Spectrum Shift: The algorithm correctly identified that omitting a declared mode leads to failure; it only recovered performance when the new mode was explicitly declared.
  • Learned Mesh Proposer: When applied to meshes generated by a Fourier Neural Operator (FNO), the pre-solve audit (after generator-specific calibration) successfully detected rare scale-structured failures that the FNO missed, acting as a high-specificity sentinel.

Significance and Claims
The paper modestly claims that this method is a scale-sensitive, certificate-first screen for specific scalar classes, not a universal error estimator.

• It does not claim to replace layer-adapted mesh generation or residual estimation.
• It does not claim to discover undeclared modes; the working spectrum must be declared a priori.
• The primary contribution is the algorithmic workflow that combines a deterministic, scale-covariant audit with a risk-calibrated decision process.
• The method is particularly valuable for scenarios where a reference solution is unavailable, offering a way to certify safety or defer to refinement with quantified risk, leveraging the geometric properties of multiplicative characters on positive meshes.