Generalized gaussian bounds for discrete convolution powers

基于 Sturm-Liouville 理论的圆盘上卷积的离散谱

Discrete Spectra of Convolutions on Disks using Sturm-Liouville Theory

本文基于 Sturm-Liouville 理论系统研究了圆盘上函数卷积的离散谱方法，并探讨了零值边界条件和导数边界条件下的理论表现。

关于在半轴上求解时间依赖的薛定谔方程的离散和半离散透明边界条件的一些评论

Some remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

本文研究了在半轴上求解时间依赖薛定谔方程的离散和半离散透明边界条件，证明了两种TBC卷积核的差值界，并通过数值实验验证了半离散TBC的有效性。

迭代改进的Lavrentiev正则化方法用于离散亥姆霍兹型微分滤波器的反卷积

Iterative Refinement of A Modified Lavrentiev Regularization Method for De-convolution of the Discrete Helmholtz Type Differential Filter

本文提出并分析了改进的Lavrentiev正则化方法用于离散亥姆霍兹型微分滤波器的反卷积，通过利用亥姆霍兹滤波器的特性，改进了正则化方法，降低了近似解与原解之间的误差界，并推导了最优停止条件。

Convergence of Multilevel Stationary Gaussian Convolution

It is well-known that polynomial reproduction is not possible when approximating with Gaussian kernels. Quasi-interpolation schemes have been developed which use a finite number of Gaussians at different scales, which then reproduce polynomials of low degree \cite{beatson}, and thus achieve polynomial orders of convergence. At the same time, interpolation with kernels of fixed width suffers from an explosion in condition number, and information from all data points influences the approximation at any one data point (no localisation). In \cite{HL1} the authors show that, for periodic convolutio…

Convergence of sparse grid Gaussian convolution approximation for multi-dimensional periodic function

We consider the problem of approximating $[0,1]^{d}$-periodic functions by convolution with a scaled Gaussian kernel. We start by establishing convergence rates to functions from periodic Sobolev spaces and we show that the saturation rate is $O(h^{2}),$ where $h$ is the scale of the Gaussian kernel. Taken from a discrete point of view, this result can be interpreted as the accuracy that can be achieved on the uniform grid with spacing $h.$ In the discrete setting, the curse of dimensionality would place severe restrictions on the computation of the approximation. For instance, a spacing of $2…

高斯曲率的离散格式及其收敛性

Discrete schemes for Gaussian curvature and their convergence

本文综述了高斯曲率的几种离散格式，提出了一种新的离散格式并证明了其在价数不小于5的正则顶点处的收敛性，同时通过反例表明价数为4时无法构造收敛的离散格式，最后比较了多种离散格式的渐近误差。

A semi-discrete numerical method for convolution-type unidirectional wave equations

Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a fini…

高斯平滑与高斯导数的离散近似

Discrete approximations of Gaussian smoothing and Gaussian derivatives

针对离散数据下高斯平滑与导数的近似问题，提出三种离散卷积方法，经理论与实验验证，极精细尺度下高斯核离散模拟方法更优，尺度参数大于约1倍网格间距时采样法的数值近似效果好。