Svenja Hüning, Johannes Wallner
This paper provides a strategy to analyse the convergence of nonlinear analogues of linear subdivision processes on the sphere. In contrast to previous work, we study the Riemannian analogue of a linear scheme on a Riemannian manifold with positive sectional curvature. Our result can be applied to all general subdivision schemes without any sign restriction on the mask.
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Analysis and convergence of Hermite subdivision schemes
本文通过研究向量细分算子并建立Hermite细分格式、向量级联算法与可细化向量函数之间的关系，解决了Hermite掩模刻画、矩阵值掩模因式分解及收敛性等关键问题。
Convergence and normal continuity analysis of non-stationary subdivision schemes near extraordinary vertices and faces
本文针对具有任意拓扑结构的2维流形网格，研究非平稳细分方案的收敛性和法向连续性，提出新的充分条件以保证在奇异顶点和面附近的收敛性和法向连续性。
Improving the convergence analysis of linear subdivision schemes
本文研究二元单变量线性细分格式的收敛性，证明收敛格式的收缩因子不低于1/2，从而说明样条类格式具有最优收敛率，并给出新的收敛条件及改进判定算法。
Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature
研究流形上非正截面曲率下的细分方案的well-defined性和收敛性，通过将仿射平均替换为黎曼中心质点，分析其Hölder连续性及收敛性条件。
This paper presents an enhanced version of our previous work, hybrid non-uniform subdivision surfaces [19], to achieve optimal convergence rates in isogeometric analysis. We introduce a parameter $λ$ ($\frac{1}{4}<λ<1$) to control the rate of shrinkage of irregular regions, so the method is called tuned hybrid non-uniform subdivision (tHNUS). Our previous work corresponds to the case when $λ=\frac{1}{2}$. While introducing $λ$ in hybrid subdivision significantly complicates the theoretical proof of $G^1$ continuity around extraordinary vertices, reducing $λ$ can recover the optimal convergence…
Multivariate Vector Subdivision Schemes with a General Matrix-valued Filter
该研究针对通用矩阵值滤波器的向量细分格式收敛性不明确的问题，从向量级联算法角度给出唯一合理的向量细分格式定义，证明其收敛性与对应级联算法等价，可推导改进已有细分格式收敛结果并给出示例。
Subdivision schemes based on weighted local polynomial regression. A new technique for the convergence analysis
针对计算机辅助设计中含噪数据生成曲线曲面的问题，本文提出基于加权局部多项式回归的二元单变量线性细分格式家族，研究其收敛性等性质并开发新理论结果，通过例子验证性质。
Vector Subdivision Schemes for Arbitrary Matrix Masks
本文针对任意矩阵掩模定义向量细分格式，证明其收敛性等价于关联向量级联算法，并研究收敛速率，从而推广了Lagrange和Hermite细分格式的现有理论。
Convergence of univariate non-stationary subdivision schemes via asymptotical similarity
研究提出渐近相似性概念，用于非平稳细分方案的收敛性分析，放宽了渐近等价条件，适用于多种重要非平稳方案。
Manifold-valued subdivision schemes based on geodesic inductive averaging
本文提出基于测地诱导平均的流形值细分方案，通过适应线性细分方案实现收敛性保证，适用于完全测地流形。
Point-Normal Subdivision Curves and Surfaces
本文提出点-法向细分方案，通过联合细化顶点位置和单位法向，将线性细分推广到非线性，能重现圆、圆柱和球面，保持收敛性与平滑阶，并生成视觉$C^2$曲面，实验验证了其有效性。
Convergence of Subdivision Schemes and Smoothness of Refinable Functions on p-adic Fields
本文系统研究了p-adic域上的加细方程与细分格式，用q-范数联合谱半径刻画Lq收敛性，并分析了Qp上复值函数的光滑性。
Anisotropic, interpolatory subdivision and multigrid
本文提出适用于各向异性多网格方法的多变量网格传输算子家族，基于新的各向异性插值细分方案，研究其最小性、多项式再生和收敛性质，并与对应多网格方法的收敛性和最优性联系起来。
In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order $d$ allows for $d$ factorizations of the subdivision operator with respect to the Gregory operators: A new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the $d$-th factorization provides a ``convergence from contractivity'' method for showing $C^d$-convergence of the associated Hermite subdivision scheme. The power of …
Analysis of a class of non linear subdivision schemes and associated multi-resolution transforms
本文研究一类定义为线性细分格式扰动的非线性细分格式的收敛性和稳定性，并应用于多种具体格式。
An isogeometric approach for solving the Laplace-Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull-Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace-Beltrami equation requires only C0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull-Clark subdivision surfaces. Catmull-Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry wi…
Multigrid methods: grid transfer operators and subdivision schemes
本文研究多网格方法中网格传输算子的新定义与分析，通过关联粗网格修正的符号分析与单变量细分方案的逼近特性，提出新的高阶问题适用的网格传输算子，并展示其在多网格方法中的收敛性和最优性。
Provably convergent implementations of the subdivision algorithm for the computation of invariant objects
本文改进了子划分算法，证明在弱假设下算法收敛，无需精细采样或高精度动力学模拟。
Subdivision Shell Elements with Anisotropic Growth
提出一种基于Loop细分曲面的薄壳有限元方法，用于处理大变形和各向异性生长，通过数值模拟展示其广泛应用。
Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P estimates the geodesic curvature of C. This result allows us to evaluate the geodesic curvature of discrete curves on surfaces. In particular, we apply such result to planar and spherical 4-point angle-based subdivision …
Generalized Taylor operators and Hermite subdivision schemes
本文提出广义泰勒算子，保留原有优势，使赫尔米特细分方案收敛性理论得以扩展，包含卡迪纳样条并构造新类型收敛方案。
Smoothness of Nonlinear and Non-Separable Subdivision Schemes
本文研究多元非线性细分格式，通过差分格式的压缩性条件，证明极限函数在Lp和Sobolev空间中的正则性。
Multivariate Generalized Hermite Subdivision Schemes
本文提出广义 Hermite 细分格式，刻画其收敛性、光滑性与矩阵掩模性质，并构造性地证明对任意正整数 m 存在具有线性相位矩的 $C^m$ 光滑样条基向量函数，统一解决了 Lagrange、Hermite 和 Birkhoff 细分格式的存在性与收敛性问题。
Subdivision Schemes in Metric Spaces
提出完整度量空间上非线性细分格式的统一框架，通过收缩性和两种邻近性概念证明收敛定理，并应用于紧集度量空间、Wasserstein空间等实例。
Interpolating Refinable Functions and $n_s$-step Interpolatory Subdivision Schemes
本文提出并研究了 $n_s$ 步插值 $M$-细分格式及其插值 $M$-细化函数，完全用掩模刻画其收敛性与光滑性，并引入 $r$-掩模拟平稳细分格式，通过结合两者构造更高步数的插值细分格式，给出 $M=2,3,4$ 的实例。
Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes
本文基于多相表示，针对满足Strang-Fix条件的多元三角多项式提出一种构造性分解方法，并应用于任意膨胀矩阵下多变量细分格式的收敛性与光滑性分析，给出了充分条件。
Multiple multivariate subdivision schemes: matrix and operator approaches
本文扩展了基于矩阵的方法，用于研究多重细分方案。通过分析多重细分方案的收敛性，利用子细分权重导出的平方矩阵的联合谱半径来表征其收敛性。
Quasi-stationary Subdivision Schemes in Arbitrary Dimensions
针对平稳细分格式提升光滑性需扩大掩模支撑域带来的实现复杂等问题，本文提出多元准平稳细分格式，刻画其收敛与光滑性，给出短支撑插值掩模设计方法，并通过二进膨胀网格实例验证其可规避平稳格式的相关难题。
Nonlinear stationary subdivision schemes that reproduce trigonometric functions
本文定义了一类可再现圆锥曲线形状（含至多二阶多项式）的非线性平稳插值细分格式，相比需不同细化规则的线性非平稳格式，该格式用同一规则即可精确再现不同圆锥曲线，并分析了其收敛性等性质及获得$\mathcal{C}^1$极限函数的条件。
A uniform non-linear subdivision scheme reproducing polynomials at any non-uniform grid
本文提出一种非线性统一细分格式，可在非均匀网格上再现二次多项式数据且无需网格先验知识，通过零化算子推断网格，以非平稳方式定义并证明收敛性，数值例子验证其生成曲率连续曲线的效用。
Weighted least squares subdivision schemes for noisy data on triangular meshes
本文提出了一种加权最小二乘细分方案，用于处理三角网格上的噪声数据，通过局部拟合多项式实现去噪和高精度逼近。
On a C2-nonlinear subdivision scheme avoiding Gibbs oscillations
提出一种新的非线性细分格式以消除间断附近的吉布斯振荡，并证明其收敛性、稳定性和逼近阶，极限函数具有大于1.192的Hölder正则性指数。
Univariate subdivision schemes for noisy data
本文提出并分析了用于细化噪声数据的单变量线性平稳细分方案，通过局部最小二乘多项式拟合。研究了这些方案的收敛性、光滑性和极限函数，并通过数值实验展示了其效果。
Limits of level and parameter dependent subdivision schemes: a matrix approach
本文提出一种矩阵方法分析依赖参数的细分方案收敛性及Hölder正则性，通过联合谱半径方法高效判断，并通过实例展示参数依赖的重要性及方案的限制。
Scalar Subdivision Schemes and Box Splines
本文利用Möller和Sauer的结果，通过证明掩码符号必须是某些商多项式理想中平移箱样条生成元的线性组合，刻画了满足k阶和规则的标量d元细分格式（膨胀矩阵2I）的结构，并指出箱样条细分是多元细分格式的基本构建块。
Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions
本文推广了计算联合谱半径的不变多面体算法，并将其扩展到更广泛的矩阵集，用于解决小波和细分的正则性问题。
This paper discusses the generation of multivariate $C^\infty$ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called \emph{Up-function}, by a non-stationary scheme based on masks of {spline subdivision schemes} of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes …
Complex Eigenvalues for Binary Subdivision Schemes
本文证明了具有回文掩模的收敛细分方案何时会出现复特征值，并给出了掩模大小的下限，找到了达到该下限的方案，并从几何角度解释了为何此类方案尚未用于计算机辅助几何设计。
Ellipse-preserving Hermite interpolation and subdivision
本文提出了一类分段指数函数，具有赫尔米特插值性质，用于曲线点与切线的联合插值，并能完美再现椭圆。通过格林函数方法分析其多分辨率和逼近性质，推导出可快速实现的向量和标量细分方案，其渐进行为与经典三次赫尔米特样条算法相同，保证了四阶逼近精度。
Approximation order and approximate sum rules in subdivision
本文研究非平稳细分方案中指数多项式和近似求和规则的作用，证明了其与平稳情况中多项式和求和规则的类比关系，包括指数多项式的再生与近似求和规则的关系，以及近似阶的确定。
High-order complete flux schemes for convection-diffusion equations on arbitrary subdivisions
研究对流扩散方程在任意网格的高阶格式，通过推导精确法向通量构建完全通量有限体积法，对任意多项式次数离散空间有效，在不同空间有良好表现，数值实验验证了方法的稳健性与最优精度。
Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials
本文研究了指数伪样条的生成与再现特性，推导了其子分割符号的显式表达，并分析了其对称性、收敛性和正则性。
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average - a notio…
We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge-Ampère type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge-Ampère type equations on the plane. Existing techniques for proving convergence do not immediately app…
In this paper, we give pointwise estimates of a Voronoï-based finite volume approximation of the Laplace-Beltrami operator on Voronoï-Delaunay decompositions of the sphere. These estimates are the basis for a local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient. Here, we consider the Voronoï-based finite volume method as a perturbation of the finite element method. Finally, using regularized Green's functions, we derive quasi-optimal convergence order in the maximum-norm with minimal regularity requirements. Numerical examples show tha…
Convergence properties of Donaldson's $T$-iterations on the Riemann sphere
本文通过黎曼球面和高维复射影空间$\mathbb{CP}^n$的实例，研究Donaldson定义的$T, T_ν, T_K$算子的迭代收敛到平衡度量并逼近常标量曲率度量的性质。
Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere
研究离散全纯函数在黎曼球面分支覆盖中的收敛性，通过三角剖分和立体投影定义离散调和函数，并证明离散周期矩阵收敛于连续情形，同时给出误差估计和离散全纯积分的收敛结果。
Euler-Maruyama approximations of the stochastic heat equation on the sphere
针对球面上加性各向同性Wiener噪声驱动的随机热方程，提出空间谱方法结合时间Euler-Maruyama格式的逼近方案，推导了最优强收敛阶及二阶矩收敛阶，并通过数值模拟验证了理论结果。
Numerical analysis of lognormal diffusions on the sphere
本文研究了单位球面上静止扩散方程的数值解，通过获得各向同性高斯随机场的Hölder正则性，将其与驱动的对数正态系数的正则性联系起来，从而在Sobolev空间中获得解的L^p正则性。随后推导了多级蒙特卡洛有限元和谱元离散化方法的收敛率估计，并通过数值实验验证了理论结果。
We investigate the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a functi…
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