Local Regularity Estimation through Sobolev-Scale Norm Profile

TL;DR

This paper introduces a kernel-based method to estimate the local Sobolev regularity of functions from scattered data by analyzing the growth of Sobolev norms of interpolants, enabling accurate spatial regularity profiling.

Contribution

The paper presents a novel Sobolev-scale norm profile approach for local regularity estimation, incorporating strategies to improve robustness and accuracy in kernel-based function analysis.

Findings

Accurately estimates local Sobolev regularity from scattered data.

Demonstrates robustness on synthetic and turbulent-flow datasets.

Provides a quantitative tool for analyzing spatially varying smoothness.

Abstract

We develop a kernel-based approach for estimating the spatially varying Sobolev regularity~$s$ of an unknown $d$-variate function~$f$ from scattered sampling data, which quantifies the degree of local differentiability supported by the data. Relying only on neighborhood data near the point of interest $z\in \Omega_z$, our method constructs a sequence of Sobolev-space reproducing kernel interpolants whose kernel smoothness order is specified by an index~$m > d/2$. The native-space norms of these interpolants are evaluated over a bounded range of~$m$, producing a \emph{Sobolev-scale norm profile}. The elbow of this profile serves as a quantitative probe of the underlying local regularity~$s(\Omega_z)$. In particular, when $m > s(\Omega_z)$, the profile exhibits rapid, near-worst-case growth governed by the classical upper bound associated with the conditioning of the kernel matrix. A band-limited surrogate analysis explains this transition and establishes a lower-bound relation linking native-norm growth to the Sobolev regularity of~$f$. Two complementary strategies are incorporated for further enhancement: (i)~a \emph{stencil-shift} subroutine, which repositions local neighborhoods to avoid crossing discontinuities whenever possible, thereby suppressing artifacts in the norm estimates; and (ii)~a local--global \emph{norm-sweep comparison} strategy that combines short two-point local tails with an optional one-point global screen to detect outlier $\Omega_z$ of low Sobolev regularity and accelerate evaluation on large datasets. Numerical experiments on synthetic test functions and turbulent-flow data demonstrate accurate recovery of spatially varying regularity and confirm the robustness of the proposed characterization for kernel-based approximation and differentiation.