亚纯函数的微分矩阵

球面上核微分矩阵的谱稳定性与扰动结果

Spectral stability and perturbation results for kernel differentiation matrices on the sphere

本文研究球面上由配点生成的核微分矩阵的谱性质，给出全局矩阵的扰动估计并证明谱符号保持，进而推导局部RBF-FD矩阵的谱界，数值实验验证理论结果。

高效求解参数依赖的拟分离系统及解析矩阵函数的计算

Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions

本文提出利用拟分离结构在对角移位和倒置下的不变性，高效求解参数依赖的矩阵方程，并通过数值实验验证方法的快速性和数值稳定性。

极化差分矩阵用于盘上满足非齐次狄利克雷、Neumann和Robin边界条件的拉普拉斯方程及满足非齐次狄利克雷条件的双调和方程

Polar differentiation matrices for the Laplace equation in the disk subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions

本文提出了一种在盘上的伪谱方法，不同于以往的方法，盘不需要重复。通过仅使用相应的差分矩阵元素，解决了满足非齐次狄利克雷、Neumann和Robin边界条件的拉普拉斯方程以及满足非齐次狄利克雷条件的双调和方程。值得注意的是，该方法不使用任何四阶积分，不需求解任何解耦的常微分方程组，不使用任何极点条件，也不需要任何提升。通过几个数值示例展示了谱收敛性。本文提出的伪谱方法可用于估计Sherwood数，将质量通量整合到盘上，并可轻松应用于求解Lotka-Volterra系统和涉及化学反应的非线性问题。

A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

In this paper we explore orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for $\mathrm{L}_2(\mathbb{R})$, have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this famil…

Automatic differentiation for Riemannian optimization on low-rank matrix and tensor-train manifolds

In scientific computing and machine learning applications, matrices and more general multidimensional arrays (tensors) can often be approximated with the help of low-rank decompositions. Since matrices and tensors of fixed rank form smooth Riemannian manifolds, one of the popular tools for finding low-rank approximations is to use Riemannian optimization. Nevertheless, efficient implementation of Riemannian gradients and Hessians, required in Riemannian optimization algorithms, can be a nontrivial task in practice. Moreover, in some cases, analytic formulas are not even available. In this pape…

线性代数函数的算法微分及其在最优实验设计中的应用（扩展版）

Algorithmic Differentiation of Linear Algebra Functions with Application in Optimum Experimental Design (Extended Version)

本文推导了矩形QR分解和对称矩阵特征值分解的高阶导数算法，并应用于最优实验设计中目标函数的梯度计算。

基于多区间 Fourier 延拓的快速数值导数

Fast numerical derivatives based on multi-interval Fourier extension

针对有界区间含噪均匀采样数据的稳定数值微分问题，提出结合多区间 Fourier 延拓与自适应递归二分的快速算法，显著提升导数重构精度与计算效率。