发表机构
School of Big Data and Statistics, Anhui University（安徽大学大数据与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出统一向量拟插值理论，通过频域正交投影构造矩阵核，精确保持散度自由、无旋等物理约束，并引入广义Helmholtz-Hodge分解，实现无约束优化的保结构逼近与误差估计。
AI 中文摘要
我们发展了一个向量拟插值的统一理论，该理论精确保持由线性常系数微分算子系统刻画的固有物理结构，包括散度自由和无旋约束作为特例。关键要素是一族由频域中的正交投影算子构造的矩阵值核，其列解析地满足指定的约束。所得的拟插值是核平移与采样函数值作为系数的加权组合，因此不需要约束优化。在偏差-方差框架内建立了逼近函数及其导数的同时误差估计。对于数值向量场分解，我们引入了一种广义的Helmholtz-Hodge分解，将光滑向量场分解为具有不同物理结构的两个分量，并发展了相应的保结构拟插值格式和误差估计。针对几个代表性约束推导了显式核，包括稳态声学约束、组合散度-旋度约束、耦合散度自由约束以及二阶Saint-Venant相容性条件。数值实验证实了理论误差估计和精确结构保持，展示了源和汇的准确识别，并显示了在没有先验约束假设的情况下对一般光滑向量场的广义Helmholtz-Hodge分量的三维重建。
英文摘要
We develop a unified theory of vectorial quasi-interpolation that exactly preserves intrinsic physical structures characterized by systems of linear constant-coefficient differential operators, including divergence-free and curl-free constraints as special cases. The key ingredient is a family of matrix-valued kernels constructed from an orthogonal projector in the frequency domain, whose columns analytically satisfy the prescribed constraints. The resulting quasi-interpolant is a weighted combination of kernel translates with sampled function values as coefficients, and therefore requires no constrained optimization. Simultaneous error estimates for the approximand and its derivatives are established within a bias-variance framework. For numerical vector-field decomposition, we introduce a generalized Helmholtz--Hodge decomposition that splits a smooth vector field into two components with distinct physical structures, and develop corresponding structure-preserving quasi-interpolation schemes and error estimates. Explicit kernels are derived for several representative constraints, including steady-state acoustic constraints, combined divergence-curl constraints, coupled divergence-free constraints, and second-order Saint-Venant compatibility conditions. Numerical experiments confirm the theoretical error estimates and exact structure preservation, demonstrate accurate identification of sources and sinks, and show three-dimensional reconstruction of generalized Helmholtz--Hodge components for general smooth vector fields without prior constraint assumptions.
