Local Regularity Estimation through Sobolev-Scale Norm Profile

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.