Local Regularity Estimation through Sobolev-Scale Norm Profile

TL;DR

The paper addresses the problem of estimating spatially varying Sobolev regularity from scattered data by introducing a kernel-based Sobolev-scale norm profile. By sweeping a family of Sobolev-space kernels with increasing smoothness $m>d/2$ and evaluating the native-norm $\|u_m\|_{\mathcal{N}_{\Phi_m}}$ on local stencils, it defines an elbow-based estimator for the local Sobolev order $s(z)$ via $\tilde{s}(\Omega_z)=m^*$, where $m^*$ marks a transition in $\log\eta(m)$. A band-limited surrogate analysis provides a rigorous lower bound linking the growth of $\eta(m)$ to the underlying regularity through a spectral median $\beta_t(f_\sigma)$, establishing a theoretical bridge between native norms and Sobolev smoothness. The framework is augmented with practical extensions: stencil-shift to sharpen localization and a fast two-point norm-sweep strategy to scale to large, nonuniform datasets. Numerical experiments on synthetic and turbulent-flow data demonstrate accurate recovery of spatially varying regularity and robust performance for regularity-aware differentiation, with clear implications for adaptive stencils and PDE solvers on scattered grids.

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 $z\in Ω_z$, 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~$s(Ω_z)$. In particular, when $m > s(Ω_z)$, 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 $Ω_z$ 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.

Figures (17)

• Figure 1: One-dimensional RBF interpolation on local stencils using Whittle--Matérn kernels with smoothness $m=1,3,5$. The test function is $f(x)=\sin(2\pi x)$ for $x>0$, $f(x)=0$ for $-0.5<x\le0$, and $f(x)=1$ for $-1\le x\le-0.5$. A total of 51 sample points are used, and each local interpolant is computed on a stencil of 10 points.
• Figure 1: Band-limited kernel interpolation of the one-dimensional step function $f(x)=\chi_{(0,\infty)}(x)$ from uniformly spaced samples. The interpolants $u_{\sigma,t}$ use Fourier-domain kernels $\widehat{\Phi}_{\sigma,t}(\omega)=(1+\omega^2)^{-t}\,\mathbf{1}_{|\omega|\le\sigma}$, with $t=1,\ldots,5$. The Fourier-truncated kernel is used here solely to illustrate the band-limited surrogate viewpoint and the stabilization of the interpolant as $t$ increases.
• Figure 1: (a) A symmetric stencil spanning a discontinuity produces an underestimated data-driven regularity $\tilde{s}(\Omega_z)$. Shifting the stencil slightly away from the jump avoids mixed samples, leading to a higher $\tilde{s}(\Omega_z)$ and sharper localization of the smooth region. (b) The test function is $f(z)=\tfrac{1}{2}z^{2}\mathbf{1}_{{z\ge 0}}$ and the target location is $z = 0$, taken as an off-sample point ($z\notin X$). Different shifted neighborhoods yield markedly different norm profiles $\eta(m)$. In the extreme case where the shifted stencil becomes highly one-sided and the center approaches the stencil boundary (red), the profile can appear artificially mild, leading to an overestimation of the local regularity $\tilde{s}$ under Definition \ref{['def:data-driven-regularity']}. This motivates the interior admissibility constraint used in the stencil-shift procedure.
• Figure 1: (Example 1) Discrete native-norm profiles $\eta(m)$ for three one-dimensional test functions (a step function, a first-order kink, and a second-order kink), corresponding to Sobolev regularities $H^{0.5-}$, $H^{1.5-}$, and $H^{2.5-}$, respectively. A Whittle--Matérn kernel with $\varepsilon=1$ is used on $\Omega=[-0.5, 0.5]$. Each curve corresponds to a different stencil size $n$ ranging from $5$ to $50$.
• Figure 2: Native‑norm profiles $\eta(m)$ versus kernel smoothness $m$ for RBF interpolants centered at $z=0$, evaluated on local neighborhoods $\Omega_z=[-1,1]$ constructed from point sets with separation distances $q_X=0.1,0.05,$ and $0.025$. Colored curves correspond to a step function $f(x)=\chi_{(0,\infty)}(x)$, first- and second-order kink (given by the first and second antiderivatives $f^{(-1)}$ and $f^{(-2)}$ of $f$). Dashed and solid black lines show $\lambda_{\min}^{-1/2}(\Phi_m(X,X))$ and its approximation $(\pi/(2q_{X}))^{\,m-d/2}$, respectively, illustrating the worst-case rate in \ref{['eq:upper-bound']}.

Theorems & Definitions (8)

• Definition 2.1: Data-driven regularity
• Definition 3.1: Spectral median
• Lemma 3.2
• Proof 1
• Lemma 3.3
• Proof 2
• Theorem 3.4
• Proof 3