报告题目：Kernel-based Regularity Estimation I--Varying Smoothness
报告人： Leevan Ling （凌立云）
报告时间：2025年12月29日下午 14:00-15:00
报告地点：红瓦楼726
摘要：We consider the estimation of local regularity of an unknown function $f : \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$ from scattered samples $(X, f(X))$. When the target function $f$ is nonsmooth, standard symmetric kernel or meshfree approximations often become unstable and inaccurate. Instead of viewing this as a numerical defect, we interpret such instability as a source of diagnostic information. By employing a family of Sobolev-space reproducing kernels of varying smoothness order and examining how the corresponding native norms change across the family, we uncover how the onset of instability encodes the local smoothness of $f$. The worst-case growth of these norms as a function of Sobolev order provides a quantitative indicator of local differentiability. The underlying theoretical justification can be understood through a band-limited surrogate analysis, which establishes quantitative links between native-norm growth and the local Sobolev regularity of the data. Numerical experiments demonstrate accurate singularity detection and stable derivative approximation on scattered data, improving both robustness and accuracy near nonsmooth features at moderate additional cost.
报告人简介：Leevan Ling, Head & Professor in Department of Mathematics, Hong Kong Baptist University. He received his Ph.D. at Simon Fraser University in 2003. Prof. Ling’s research interests include scientific computing, meshless methods, and numerical analysis. He authored more than 70 publications in international journals and has joint the editorial boards of Engineering Analysis with Boundary Element (since 2008), Advances in Applied Mathematics and Mechanics (since 2010) and International Journal of Computer Mathematics (since 2020).
报告邀请人：张冉
