Qi Ye
In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels.
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Numerical cubature on scattered data by adaptive interpolation
本文提出在已知代数求积公式支集上重采样，并利用多项式、RBF和PUM自适应插值构建散乱数据求积方法，为直接无网格RBF求积和最小二乘求积提供有前景的替代。
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials
提出一种基于共轭梯度法和非均匀快速傅里叶变换的快速算法，在温和假设下实现环面上散乱数据最优插值，总复杂度为O(M log M)。
Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds
本文研究了在球面及其他流形上任意分布点上函数的Hermite-Birkhoff插值问题，提出了一种基于基函数的插值方法，该方法无需求解线性方程组，且基函数具有或thonormal性质和零导数特性。
On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions
本文提出并分析了一种结合核拟插值与多尺度技术的逼近方法，适用于散乱标量及流形值函数，证明其收敛性并展示其在去噪、异常检测等任务中的优越性能。
Error estimates for interpolation of rough data using the scattered shifts of a radial basis function
本文针对粗糙函数，利用径向基函数的散乱平移进行插值，建立了当插值点填充有界域时的误差估计。
Interpolation of scattered data in $\mathbb{R}^3$ using minimum $L_p$-norm networks, $1<p<\infty$
本文研究了在R^3中使用具有最小二阶导数L_p范数的光滑曲线网络进行散乱数据插值的极值问题，针对1<p<∞的情况给出了完整的解决方案，并通过数值实验验证了结果。
For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in euclidean spaces or spherical basis functions. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner-type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF's and to study G…
Approximate Approximations from scattered data
通过将均匀网格上的光滑快速衰减函数的伸缩平移近似拟插值扩展到散乱数据拟插值，利用多项式乘以基函数构造近似单位分解，实现指定精度的高阶逼近。
Hybrid Gaussian-cubic radial basis functions for scattered data interpolation
本文提出混合高斯-三次径向基函数以解决传统RBF在处理大数据集时的病态问题，通过粒子群优化确定最优参数，提升插值稳定性与精度。
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high dimensional data, which features bounding stoc…
Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation
本研究针对有限离散样本的多元逼近问题，通过组合低维分量核构造乘积核，分析其插值特性与张量积结构（尤其网格样本），开发牛顿基高效计算方案，数值示例验证其良好性能与灵活性。
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
研究通过限制正定核（如径向基函数）在光滑紧致嵌入子流形上构造核插值，并给出本征空间的完整刻画及Sobolev型误差估计。
With the rapid growth of data, how to extract effective information from data is one of the most fundamental problems. In this paper, based on Tikhonov regularization, we propose an effective method for reconstructing the function and its derivative from scattered data with random noise. Since the noise level is not assumed small, we will use the amount of data for reducing the random error, and use a relatively small number of knots for interpolation. An indicator function for our algorithm is constructed. It indicates where the numerical results are good or may not be good. The corresponding…
In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effe…
Approximation of a Multivariate Function of Bounded Variation from its Scattered Data
本文提出了一种从散乱数据近似有界变差函数的方法，克服了传统RBF插值方法的局限性。
Scattered data interpolation on the torus by compactly supported multinode Shepard operators
本文提出适配环面几何的紧支撑多节点谢泼德算子，结合局部多项式插值与紧支撑权重实现环面上散乱数据插值，经数值实验验证其收敛性与适用性。
Adaptive Non-Linear Partition of Unity Methods for Scattered Data Interpolation with Discontinuities
研究具有不连续性的散乱数据插值问题，核心方法是通过留一法交叉验证结合全局优化自适应调整非线性单位分解方法的超参数，主要贡献是降低不连续性附近逼近误差，在平滑区域保持精度且无需界面几何先验知识。
The dimension weighted fast multipole method for scattered data approximation
针对高维各向异性数据的散乱数据逼近问题，本文提出一种基于稀疏多项式插值退化核的维度加权快速多极子方法，可处理比标准黑盒方法更多维度，并给出严格误差估计与数值验证，成功应用于形状不确定性量化。
Kernel interpolation on generalized sparse grids
本文研究了等维和不同维乘积区域上的散乱数据逼近问题，提出基于稀疏网格组合技术的高效算法，通过改进的核插值误差估计和稀疏直接求解器处理大规模非局部再生核问题。
Multivariate Rational Approximation of Scattered Data Using the p-AAA Algorithm
本研究将原本适用于网格数据的p-AAA算法拓展至散乱多元数据场景，通过带约束的有理最小二乘优化及结构化矩阵奇异值分解实现闭式求解，验证了方法的计算特性与有效性。
Approximation with Conditionally Positive Definite Kernels on Deficient Sets
本文研究亏缺集合上条件正定核的泛函逼近，证明多项式相容性足以保证最优恢复性质，并给出拉普拉斯算子稀疏数值微分和椭圆上函数精确逼近的应用。
Fitting scattered data with optional monotonicity constraints on GPU: LipFit package
提出一种多变量散乱数据插值与逼近方法，在满足单调性约束下产生最优Lipschitz连续逼近，并实现GPU并行化的Python包LipFit。
Samplets: Wavelet concepts for scattered data
该研究提出可定义在任意维度任意分布数据点上的samplet小波型测度，将小波分析的经典概念迁移到散乱数据问题，实现矩阵压缩加速插值求解，并验证了samplet坐标下稀疏约束的有效性与高效求解能力。
RBF Interpolation with CSRBF of Large Data Sets
本文研究了大尺寸数据集和大跨度几何坐标下RBF插值的性质，探讨了计算稳定性问题。
Isoperimetric-Combinatorial Bounds for Range-Controlled Matchings and Quasi-Interpolation from Scattered Data
该研究开发介观框架分析有限点集扰动，通过施加HT组合约束证明$Y$与$Q$存在范围为$O(r)$的完美匹配，可转移积分逼近估计并锚定$Q$，还用平移不变拟插值方法获高阶估计，关键限制是HT条件及$r\leq Ch$。
Interpolation in path-connected metric spaces using splines on embedded graphs
该研究提出一种基于嵌入图的局部图基插值框架，通过变分图样条构造连续域插值函数，经对称正定矩阵流形实验验证，其性能优于kNN插值，可用于道路连通度量空间的数据插值。
Transfinite thin plate spline interpolation
本文证明在平行超平面上的周期数据满足Beppo Levi边界条件的多薄板样条插值实际上是一种薄板样条，即它最小化Duchon型泛函。
A Unified Framework for Efficient Kernel and Polynomial Interpolation
本文提出一种结合核插值与多项式插值的统一框架，通过高效数值方法提升边界条件下目标函数的插值性能。
Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data on the Sphere and Other Manifolds
本文研究了在球面及其他流形上使用Hermite-Birkhoff公式对散点数据进行插值的问题，提出了一种基于基函数的插值方法，该方法无需求解线性方程组，具有或thonormal性质和零导数特性。
Partition of Unity Interpolation on Multivariate Convex Domains
本文提出一种新的多变量插值算法，用于在凸域内高效处理散乱数据集。通过构建kd树空间划分结构，结合局部径向基函数近似和紧支撑权重函数，实现高效的统一函数插值。
Nodal Representations for Kernel-Based Multilevel Interpolation
针对散乱数据逼近，提出核方法多层插值的节点表示，显式依赖目标函数值，证明基函数指数衰减，并分析计算成本。
A Bayesian Approach for Simultaneously Radial Kernel Parameter Tuning in the Partition of Unity Method
针对径向基函数单位分解插值的参数调优问题，提出用贝叶斯优化同步搜索形状参数与半径最优值，结合无网格单位分解法降低大散乱数据量下的计算成本，实验显示相比留一交叉验证等方法可大幅缩短搜索时间。
Greedy regularized kernel interpolation
本文提出一种贪心算法用于计算稀疏核正则化插值近似，该方法在处理昂贵函数替代时具有高效性，并证明了贪心选择规则能产生渐近近似最优误差率。
We consider the extremal problem of interpolation of convex scattered data in $\mathbb{R}^3$ by smooth edge convex curve networks with minimal $L_p$-norm of the second derivative for $1<p\leq\infty$. The problem for $p=2$ was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results in (Andersson et al., 1995) and solved the problem for $1<p<\infty$. The minimum edge convex $L_p$-norm network for $1<p<\infty$ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case $1<p<\infty$ is unique for stric…
Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI
本文研究了一种基于RBF的散乱数据插值方法，通过变量缩放函数引入不连续性，分析了核函数的本原空间和误差界，并利用机器学习分类算法提取不连续性位置，通过数值实验验证了理论收敛率，并展示了在磁流体成像中的应用前景。
We investigate an interpolation/extrapolation method that, given scattered observations of the Fourier transform, approximates its inverse. The interpolation algorithm takes advantage of modelling the available data via a shape-driven interpolation based on Variably Scaled Kernels (VSKs), whose implementation is here tailored for inverse problems. The so-constructed interpolants are used as inputs for a standard iterative inversion scheme. After providing theoretical results concerning the spectrum of the VSK collocation matrix, we test the method on astrophysical imaging benchmarks.
Partition of unity interpolation using stable kernel-based techniques
本文提出了一种新的稳定且准确的近似技术，特别适用于插值大规模离散数据集。该方法利用径向基函数作为局部近似函数，并使用局部支持权重，通过计算每个PU子域的稳定基，提高了稳定性，尤其是在处理平坦核时。此外，优化的搜索过程用于构建局部稳定基，使方法更加高效。
Local spline refinement driven by fault jump estimates for scattered data approximation
本文提出基于断层跳跃估计的局部样条细化方法，通过LR B样条拟插值方案自适应调整曲面逼近，提升散乱数据在断层处的重构精度。
Divergence-free interpolation of tangential vector fields via matrix-valued kernels
该研究提出一种无需反复应用曲面微分算子的切向量场无散核插值方法，推导了显式核表示并验证了其收敛性与稳定性，可推广至非球面。
Distributed Uncertainty Quantification of Kernel Interpolation on Spheres
针对RBF核插值因不确定性关系易受噪声影响的问题，本文提出一种分布式插值方法，用于管理和量化含噪球面数据插值引发的不确定性，数值模拟表明该方法处理复杂计算环境含噪数据实用且稳健。
We consider two-stage scattered data fitting with truncated hierarchical B-splines (THB-splines) for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares variational spline approximations, exploiting a simple fairness functional to handle data distributions with a locally varying density of points. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the whole scattered data set and a suitable strategy to …
Anisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids
本文提出一种基于各向异性核的多层插值方法，在稀疏网格上逼近高维函数，推导显式表示并证明误差估计，给出三种权重策略，数值实验验证了理论。
DataTransfer: Neural network based interpolation across non-nested meshes
本文提出基于神经网络的网格间函数映射模型，通过数据驱动回归解决插值问题，验证了径向基函数网络在精度、效率和鲁棒性上的平衡。
Optimal selection of local approximants in RBF-PU interpolation
本文研究了RBF-PU插值中局部近似量的选择问题，通过最小化先验误差估计来优化局部子域中的形状参数和线性系统规模，提升处理非均匀数据精度的能力。
Error estimates for vector field interpolation based on generalized matrix-valued kernels
本文提出算子框架构造无散无旋矩阵值核，降低势方法正则性要求，完成核插值问题的精确误差分析与稳定性分析，数值实验验证了理论结果。
A Modified Hermite Radial Basis Function for Accurate Interpolation
针对HRBF插值在低形状参数下病态导致精度下降的问题，提出引入多项式项的MHRBF方法，在保持或降低计算成本的同时提升精度，并在所有参数和域大小下实现更低误差。
The Penalized Lebesgue Constant for Surface Spline Interpolation
针对非均匀分布数据，利用类似流形上准均匀数据插值稳定性的技术，证明了曲面样条插值在Rd上具有更强的局部稳定性，并得到拉格朗日函数快速衰减的点态估计，结合勒贝格引理表明其收敛率与局部逼近方案相同。
Scattered Data Histopolation in Averaging Kernel Hilbert Spaces
本文提出在平均核希尔伯特空间中进行散乱数据插值的方法，通过构建平均核并分析其性质，提供了一种更灵活的插值框架，并通过实验验证了其有效性。
Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects
本文提出一种基于有理径向基函数的分区为一方法，用于高效准确地近似3D对象，通过局部方案提高计算效率。
This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, do…
An application of numerical differentiation formulas to discontinuity curve detection from irregularly sampled data
本文提出了一种从散乱数据中检测断点曲线（通常称为断层）的方法，通过数值微分公式近似梯度，提取靠近断点的点集，并利用局部回归线和二次最小二乘逼近重构断点形状，最后通过曲线插值算法完成重建。
The high-level error bound for shifted surface spline interpolation
针对平移表面样条插值，提出一种改进的指数衰减误差界O(ω^{1/d})，优于传统多项式阶误差界O(d^l)。
Kernel approximation beyond the native space -- with applications to approximation on manifolds
针对嵌入流形上的正定和条件正定核，提出一种推广的逼近方案，通过积分算子范围刻画目标函数，并建立新的Bernstein不等式以实现原生空间紧嵌入Sobolev空间中的插值误差估计。
Data dependent Shepard approximation through and adaptive modification of the shape parameter
提出一种数据依赖的Shepard插值方法，通过自适应调整形状参数减少一维和二维数据中跳跃间断附近的模糊，理论证明并数值验证了其有效性。
General superconvergence for kernel-based approximation
该研究针对目标函数比再生核希尔伯特空间函数更光滑的超收敛情形，刻画了一般希尔伯特空间中投影的超收敛性，拓展了相关理论，并通过数值实验验证了结果。
A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations
本文提出一种基于拉普拉斯-贝尔特拉米算子和沃罗诺伊细分的数据驱动插值方法，通过扩散过程和数据内在几何构造连续扩展，实现高效插值并提升可扩展性。
Efficient computation of partition of unity interpolants through a block-based searching technique
本文提出一种高效插值工具，适用于大规模散乱数据集。通过将径向基函数作为局部近似函数，并使用局部支持权重函数，结合基于块的空间划分数据结构，优化了最近邻点搜索过程，通过二维和三维插值的复杂度分析和数值实验验证了方法的有效性，并讨论了用于几何建模的应用及Matlab软件包。
Several engineering applications involve complex materials with significant and discontinuous variations in thermophysical properties. These include materials for thermal storage, biological tissues with blood capillaries, etc. For such applications, numerical simulations must exercise care in not smearing the interfaces by interpolating variables across the interfaces of the subdomains. In this paper, we describe a high accuracy meshless method that uses domain decomposition and cloud-based interpolation of scattered data to solve the heat conduction equation in such situations. The polyharmo…
Superconvergence of Kernel-Based Interpolation
研究核基插值的超收敛性，通过抽象理论和边界条件分析，揭示了光滑性和局部化条件对超收敛的影响，提出 Mercer 展开在超收敛中的作用。
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