arXiv subjects

Leevan Ling

Publications and source records attributed to Leevan Ling.

At least 19 recordsLinked to original sources

arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗

arXiv2026

Making Gaussian Kolmogorov-Arnold Networks Reliable and Accurate

Kolmogorov-Arnold Networks (KANs) replace fixed activations with learnable univariate edge functions whose behavior depends strongly on the chosen basis. Gaussian radial basis functions provide a simple and efficient alternative to splines, but their accuracy and stability are highly sensitive to the scale parameter $ε$, which has not been studied systematically. We analyze this dependence through the geometry and conditioning of the first-layer feature matrix. Because the first layer is defined directly on the input domain, any loss of feature distinguishability introduced there propagates through the entire network. Based on this analysis, we identify the practical operating interval \[ ε\in \left[\frac{1}{G-1},\frac{2}{G-1}\right], \] where $G$ is the number of Gaussian centers. This interval is proposed as a stable design rule rather than a universal optimum. Extensive experiments on function approximation and physics-informed problems confirm its reliability across different collocation densities, grid resolutions, architectures, and input dimensions. We also show how the same principle supports fixed-scale selection, variable-scale models, constrained optimization of $ε$, and efficient scale search using early-stage training error. These results establish scale selection as a central design principle for reliable and accurate Gaussian KANs.

cs.CE↗

arXiv2026

Local Regularity Estimation through Sobolev-Scale Norm Profile

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 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.

math.NA↗

arXiv2026

Error estimates for vector field interpolation based on generalized matrix-valued kernels

Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.

math.NA↗

arXiv2026

Trajectory-Based RBF Collocation Method via Closest-Point Embedding for Surface Advection-Diffusion Equations

We introduce a Trajectory-Based RBF Collocation (TBRBF) method for solving surface advection-diffusion equations on smooth, compact manifolds. TBRBF decouples advection and diffusion by applying a characteristic treatment with a Kansa-type RBF collocation method for diffusion PDE, which yields an operator-split characteristic (OSC) system comprising a characteristic ODE and a diffusion PDE. We rigorously prove the equivalence between the OSC system and the original surface PDE on manifolds by embedding the latter into a narrow band domain through the closest point mapping and its constant-along-normal extension. Using an intrinsic approach, we construct a time-continuous embedded PDE with push-forward operators in each chart of the atlas and establish its equivalence with the OSC system in the narrow band. Restricting the solution back to the manifold recovers the OSC system on manifolds, ensuring that the method introduces no operator splitting error. After the surface OSC system is obtained, it admits both extrinsic and intrinsic discretizations. Extensive numerical experiments confirm the robust stability and accuracy of the proposed method.

math.NA↗

arXiv2026

A Practitioner's Guide to Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs), whose design is inspired-rather than dictated-by the Kolmogorov superposition theorem, have emerged as a structured alternative to MLPs. This review provides a systematic and comprehensive overview of the rapidly expanding KAN literature. The review is organized around three core themes: (i) clarifying the relationships between KANs and Kolmogorov superposition theory (KST), MLPs, and classical kernel methods; (ii) analyzing basis functions as a central design axis; and (iii) summarizing recent advances in accuracy, efficiency, regularization, and convergence. Finally, we provide a practical "Choose-Your-KAN" guide and outline open research challenges and future directions. The accompanying GitHub repository serves as a structured reference for ongoing KAN research.

cs.LG↗

arXiv2026

Minimum-norm interpolation for unknown surface reconstruction

We study algorithms to estimate geometric properties of raw point cloud data through implicit surface representations. Given that any level-set function with a constant level set corresponding to the surface can be used for such estimations, numerical methods need not specify a unique target function for these domain-type interpolation problems. In this paper, we focus on kernel-based interpolation by radial basis functions (RBF) and reformulate the uniquely solvable interpolation problem into a constrained optimization model. This model minimizes some user-defined norm while enforcing all interpolation conditions. To enable nontrivial feasible solutions, we propose to enhance the trial space with 1D kernel basis functions inspired by Kolmogorov-Arnold Networks (KANs). Numerical experiments demonstrate that our proposed mixed-dimensional trial space significantly improves surface reconstruction from raw point clouds. This is particularly evident in the precise estimation of surface normals, outperforming traditional RBF trial spaces including the one for Hermite interpolation. This framework not only enhances processing of raw point cloud data but also shows potential for further contributions to computational geometry. We demonstrate this with a point cloud processing example.

math.NA↗

arXiv2026

B-spline-Based ALE-MFS Framework for Evolving Domains

We develop and analyze a B-spline based arbitrary Lagrangian-Eulerian method of fundamental solutions (ALE-MFS) for curvature-driven motion of two-dimensional evolving domains. Boundary points move with the material to track the geometric flow, while interior points move within an ALE framework via a harmonic extension of the boundary velocity, computed by a meshless MFS with sources on a fixed auxiliary circle, thus avoiding volumetric meshing. Boundary normals and curvature are reconstructed by an adaptive local B-spline scheme that remains robust for strongly nonconvex shapes and large deformations. A posteriori error estimates are obtained from a hatmatrix formulation of leave-one-out cross-validation (LOOCV) for both square collocation and zero-padded least-squares systems, and are complemented by maximum principle indicators for harmonic problems. Numerical experiments on circular, star-shaped, and amoeba-like domains show that square collocation suffices for moderately complex geometries, while zero-padded least-squares significantly improves interior velocity regularity and pointwise transport accuracy for strongly nonconvex shapes, without altering the source or collocation sets. The ALE-MFS algorithm also generates high-quality moving meshes for ALE-finite element methods, with larger minimum angles and slower mesh ratio growth than classical FEM mesh-motion strategies, suggesting a practical and easily integrable alternative for challenging moving-interface simulations.

math.NA↗

arXiv2026

An Adaptive Lagrangian B-Spline Framework for Point Cloud Manifold Evolution

We extend our recent curve-evolution framework based on localized B-spline interpolation to present an adaptive Lagrangian framework for the geometric evolution of point-cloud data representing smooth, codimension-one surfaces in $\mathbb{R}^3$. The method constructs overlapping, localized tensor-product B-spline patches, enabling direct, meshless surface evolution from discrete samples. Within each patch, the differentiable B-spline representation yields analytic, high-order estimates of intrinsic geometric invariants, supporting curvature-driven and geometry-coupled flows. The organization of control points facilitates coherent updates of both surface samples and spline coefficients under intrinsic velocity fields. A conditioning-aware formulation of the local interpolation system, combined with a Gauss-Seidel refinement of control points, maintains interpolation quality throughout the evolution. Adaptive knot insertion and point redistribution, guided by geometric error indicators and local sampling density, preserve surface resolution and regularity during deformation. Numerical experiments demonstrate efficient and accurate reproduction of surface evolution phenomena, including mean-curvature flow, anisotropic deformations, and coupled surface-field dynamics, establishing localized B-spline methods as precise and versatile tools for dynamic manifold approximation.

math.NA↗

arXiv2025

Sobolev Algorithm for Local Smoothness Analysis (SALSA) via Sharp Direct and Inverse Statements

We extend sharp direct and inverse approximation statements for kernel-based methods for finitely smooth kernels, i.e. those whose native spaces are norm-equivalent to Sobolev spaces. In particular, our inverse results are now formulated for a broad class of approximation schemes beyond interpolation, extending existing theory. Building on these results, we propose a novel Sobolev Algorithm for Local Smoothness Analysis (SALSA) for detecting local smoothness properties of target data, including their degree of smoothness and non-smoothness. The method is rigorously grounded based on the sharp direct and inverse statements. Numerical experiments in various settings highlight the effectiveness of the proposed algorithm.

math.NA↗

arXiv2025

Energy-conserving Kansa methods for Hamiltonian wave equations

We introduce a fast, constrained meshfree solver designed specifically to inherit energy conservation (EC) in second-order time-dependent Hamiltonian wave equations. For discretization, we adopt the Kansa method, also known as the kernel-based collocation method, combined with time-stepping. This approach ensures that the critical structural feature of energy conservation is maintained over time by embedding a quadratic constraint into the definition of the numerical solution. To address the computational challenges posed by the nonlinearity in the Hamiltonian wave equations and the EC constraint, we propose a fast iterative solver based on the Newton method with successive linearization. This novel solver significantly accelerates the computation, making the method highly effective for practical applications. Numerical comparisons with the traditional secant methods highlight the competitive performance of our scheme. These results demonstrate that our method not only conserves the energy but also offers a promising new direction for solving Hamiltonian wave equations more efficiently. While we focus on the Kansa method and corresponding convergence theories in this study, the proposed solver is based solely on linear algebra techniques and has the potential to be applied to EC constrained optimization problems arising from other PDE discretization methods.

math.NA↗

arXiv2025

Efficient manifold evolution algorithm using adaptive B-Spline interpolation

This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction--diffusion systems for pattern formation.

math.NA↗

arXiv2024

Structure-preserving Kernel-based methods for solving dissipative PDEs on surfaces

In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties.

math.NA↗

arXiv2024

Proving the stability estimates of variational least-squares Kernel-Based methods

Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.

math.NA↗

arXiv2024

Greedy Trial Subspace Selection in Meshfree Time-Stepping Scheme with Applications in Coupled Bulk-Surface Pattern Formations

Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.

math.NA↗

arXiv2024

Learning PDEs from data on closed surfaces with sparse optimization

The discovery of underlying surface partial differential equation (PDE) from observational data has significant implications across various fields, bridging the gap between theory and observation, enhancing our understanding of complex systems, and providing valuable tools and insights for applications. In this paper, we propose a novel approach, termed physical-informed sparse optimization (PIS), for learning surface PDEs. Our approach incorporates both $L_2$ physical-informed model loss and $L_1$ regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs. The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods. We provide practical demonstrations of the algorithms including linear and nonlinear systems. The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.

math.NA↗

arXiv2024

Realistic pattern formations on surfaces by adding arbitrary roughness

We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface {construction methods} yield comparable patterns to those {observed} in real-life animals.

math.NA↗

arXiv2024

Enhancing RBF-FD Efficiency for Highly Non-Uniform Node Distributions via Adaptivity

Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.

math.NA↗