Local Regularity Estimation through Sobolev-Scale Norm Profile
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911403970920448 |
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| author | Li, Xiaobin Ling, Leevan Sun, Yizhong |
| author_facet | Li, Xiaobin Ling, Leevan Sun, Yizhong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20207 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Regularity Estimation through Sobolev-Scale Norm Profile Li, Xiaobin Ling, Leevan Sun, Yizhong Numerical Analysis 65M60, 65M12, 65D30, 41A30 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. |
| title | Local Regularity Estimation through Sobolev-Scale Norm Profile |
| topic | Numerical Analysis 65M60, 65M12, 65D30, 41A30 |
| url | https://arxiv.org/abs/2601.20207 |