arXiv subjects

Zhengjie Sun

Publications and source records attributed to Zhengjie Sun.

12 recordsLinked to original sources

arXiv2026

Divergence-free interpolation of tangential vector fields via matrix-valued kernels

We develop and analyze a family of divergence-free kernel interpolation methods for tangential vector fields on the unit sphere. Starting from scalar radial kernels in Euclidean space, we construct tangent-valued, surface divergence-free matrix kernels without repeatedly applying surface differential operators. The construction separates the geometric enforcement of the divergence-free constraint from the choice of scalar generator and admits lower-order variants with reduced regularity requirements compared with classical potential-based methods. Using vector spherical harmonics, we derive explicit kernel representations and characterize their Fourier multipliers. We also introduce an inverse Laplace-Beltrami construction that preserves the multipliers of the underlying scalar zonal kernel. For interpolation at scattered nodes, we establish a lower bound for the smallest eigenvalue of the interpolation matrix and derive pointwise and Sobolev error estimates, including superconvergence for targets smoother than the native space. Numerical experiments corroborate the theoretical convergence and stability results, while additional examples on nonspherical surfaces illustrate the applicability of the kernel formula beyond the sphere.

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

Vector field multiplier operators and matrix-valued kernel quasi-interpolation

We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre differential equation and requires less stringent regularity conditions on the original zonal kernels. The induced integral operators are simple Fourier-Legendre multipliers that not only deliver optimal Sobolev error estimates (in terms of the scaling parameter) but also yield natural Helmholtz-Hodge decompositions on the $L_2$-tangential vector fields on $\mathbb{S}^2$. Via discretization of the underlying convolution integrals, we harvest a family of vector-valued quasi-interpolants that accomplish our approximation goal in the divergence/curl-free vector field. The quasi-interpolation algorithm is robust against noisy data. The implementation process is adaptive to human-improvision, involving neither evaluating the convolution integrals nor solving systems of linear equations. The computational efficiency and executory robustness of the quasi-interpolation algorithm stand in sharp contrast to the existing kernel-based vector field interpolation method.

math.NA↗

arXiv2026

Quasi-interpolation using generalized Gaussian kernels

This paper focuses on developing a framework for constructing quasi-interpolation with the highest achievable approximation order from generalized Gaussian kernels with the help of kernel restriction trick and periodization technique. We first demonstrate that when we restrict generalized Gaussian kernels satisfying generalized Strang-Fix conditions of order s over a torus, the corresponding restricted kernels in tensor-product forms fulfill periodic Strang-Fix conditions of the same order s. Then, based on these restricted kernels, we construct a periodic quasi-interpolant in Schoenberg's form and derive its error estimates for periodic function approximation over a torus, which reveals that our quasiinterpolant attains the highest approximation order s. Finally, using the periodization technique, we extend the periodic quasi-interpolant to its nonperiodic counterpart with the highest approximation order s for approximating a general function defined over a cube via a torus-to-cube transformation. This result stands in stark contrast to classical quasi-interpolation counterparts, which often yield much lower approximation orders than those dictated by the generalized Strang-Fix conditions of generalized Gaussian kernels. Furthermore, we propose a sparse grid counterpart for high-dimensional function approximation to alleviate the curse of dimensionality. Numerical simulations confirm that our quasi-interpolation scheme is simple and computationally efficient.

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

Monte Carlo quasi-interpolation of spherical data

We establish a deterministic and stochastic spherical quasi-interpolation framework featuring scaled zonal kernels derived from radial basis functions on the ambient Euclidean space. The method incorporates both quasi-Monte Carlo and Monte Carlo quadrature rules to construct easily computable quasi-interpolants, which provide efficient approximation to Sobolev-space functions for both clean and noisy data. To enhance the approximation power and robustness of our quasi-interpolants, we develop a multilevel method in which quasi-interpolants constructed with graded resolutions join force to reduce the error of approximation. In addition, we derive probabilistic concentration inequalities for our quasi-interpolants in pertinent stochastic settings. The construction of our quasi-interpolants does not require solving any linear system of equations. Numerical experiments show that our quasi-interpolation algorithm is more stable and robust against noise than comparable ones in the literature.

math.NA↗

arXiv2025

Spherical quasi-interpolation using scaled zonal kernels

We propose and study a new quasi-interpolation method on spheres featuring the following two-phase construction and analysis. In Phase I, we analyze and characterize a large family of zonal kernels (e.g., the spherical version of Poisson kernel, Gaussian, compactly-supported radial kernels), so that the underlying spherical convolution operators (upon the introduction of a scaling parameter) attains a high-order of approximation to target functions. In Phase II, we discretize the spherical integrals utilizing quadrature rules of optimal order to produce the final quasi-interpolants. Numerical experiments demonstrate that the new quasi-interpolation algorithm is robust and amenable to integrated as well as distributed ways of implementation. Moreover, the underlying error-analysis shows that by fine-tuning the scaling parameter in the radial kernels employed, the resulting quasi-interpolants achieve a well-balanced trade-off between approximation and sampling errors.

math.NA↗

arXiv2025

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 all orders on the lower bound and $L_2$ on the upper, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. Our theory achieves the desired form but may require slightly more smoothness on the kernel than the regular $>d/2$ assumption. 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↗

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

Quasi-interpolation for high-dimensional function approximation

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rules at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

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↗

arXiv2022

A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations

We propose a meshless conservative Galerkin method for solving Hamiltonian wave equations. We first discretize the equation in space using radial basis functions in a Galerkin-type formulation. Differ from the traditional RBF Galerkin method that directly uses nonlinear functions in its weak form, our method employs appropriate projection operators in the construction of the Galerkin equation, which will be shown to conserve global energies. Moreover, we provide a complete error analysis to the proposed discretization. We further derive the fully discretized solution by a second order average vector field scheme. We prove that the fully discretized solution preserved the discretized energy exactly. Finally, we provide some numerical examples to demonstrate the accuracy and the energy conservation.

math.NA↗