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 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 \emph{secant-based tail screening strategy} that uses two high-order norm evaluations to identify candidate low-regularity neighborhoods at reduced computational cost. Numerical experiments on synthetic test functions and turbulent-flow data demonstrate recovery of spatially varying regularity, while a surface conservation-law example illustrates the detection of evolving low-regularity regions in time-dependent PDE data.