Convergent non complete interpolatory quadrature rules

On Kosloff Tal-Ezer Least-Squares Quadrature Formulas

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the comb…

离散修正投影法用于具有光滑核的乌里松积分方程

Discrete Modified Projection Method for Urysohn Integral Equations with Smooth Kernels

本文提出离散修正投影法求解具有光滑核的乌里松积分方程，采用分段多项式空间和插值投影，分析收敛阶并给出数值结果。

A Convergent Quadrature Based Method For The Monge-Ampère Equation

We introduce an integral representation of the Monge-Ampère equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs for the Monge-Ampère equation with either Dirichlet or optimal transport boundary conditions. The use of higher-order quadrature schemes allows for substantial reduction in the component of the error that depends on the angular resolution of the finite difference stencil. This, in turn, allows for significant improvements in both stencil width and formal truncat…

Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing on…

求积作为最小二乘和极小化极大问题

Quadrature as a least-squares and minimax problem

本文通过将插值求积公式的权重向量表示为超定线性系统的最小二乘解，并建立其与极小化极大解的关系，定义了求积规则的角度等参数以评估和比较不同规则。

使用 $hp$-FEM 对时空分数阶抛物型方程的一种指数收敛离散

An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

针对时空分数阶抛物型问题，结合基于 sinc 求积的 Riesz-Dunford 积分离散与空间 $hp$-FEM，提出指数收敛格式，严格证明其在小时间及数据不兼容启动奇点下的鲁棒性。

Fast and reliable high accuracy computation of Gauss--Jacobi quadrature

Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be generally faster than previous methods and without practical restrictions on the range of the parameters. The evaluation of the nodes and weights of the quadrature is exclusively based on convergent processes which, together with the fourth order convergence of the fixed point method for computing the nodes, makes this an ideal approach for high accuracy computati…

一种求解涉及正则增生算子分数幂的稳态扩散方程的求积格式

A quadrature scheme for steady-state diffusion equations involving fractional power of regularly accretive operator

本文针对含正则增生算子分数幂的非局部扩散方程，提出一种指数收敛的求积格式，并给出覆盖多种参数情形的精确误差界，数值实验验证了估计的准确性。

On the quadrature exactness in hyperinterpolation

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree $n$ requires a positive-weight quadrature rule with exactness degree $2n$. We examine the behavior of such approximation when the required exactness degree $2n$ is relaxed to $n+k$ with $0<k\leq n$. Aided by the Marcinkiewicz--Zygmund inequality, we affirm that the $L^2$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of $n$, and this approximation scheme is convergent as $n\rightarrow\infty$ if $k$…

Fast, reliable and unrestricted iterative computation of Gauss--Hermite and Gauss--Laguerre quadratures

Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss--Hermite and Gauss--Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a conti…

A convergent numerical scheme to a parabolic equation with a nonlocal boundary condition

In this paper, a numerical scheme for a nonlinear McKendrick-von Foerster equation with diffusion in age (MV-D) with the Dirichlet boundary condition is proposed. The main idea to derive the scheme is to use the discretization based on the method of characteristics to the convection part, and the finite difference method to the rest of the terms. The nonlocal terms are dealt with the quadrature methods. As a result, an implicit scheme is obtained for the boundary value problem under consideration. The consistency and the convergence of the proposed numerical scheme is established. Moreover, nu…

用于细长体的高效收敛边界积分方法

Efficient Convergent Boundary Integral Methods for Slender Bodies

针对高纵横比纤维的粘性流体模拟中现有方法精度低或成本高的问题，提出成本与纵横比无关的高阶收敛边界积分求积方案及新型公式，实现高精度高效的细长体流体相互作用模拟。

Hölder连续、绝对收敛傅里叶级数、傅里叶余弦级数和瓦尔什级数的数值积分

Numerical integration of Hölder continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

本文提出准蒙特卡洛规则用于积分满足傅里叶、傅里叶余弦或瓦尔什系数绝对可和及Hölder条件的函数，证明积分误差收敛率为O(max((s-1)N^{-1/2}, s^{α/2}N^{-α}))。

一种收敛的有限差分-四次方程方案用于具有非局部压力的多孔介质方程

A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure

本文提出了一种新的数值方案，用于求解具有非局部压力的多孔介质方程，通过构造单调且稳定的近似方法，证明了其收敛性，并进行了数值验证。

一维浅水模型的全通量求积保正保平衡PAMPA格式

Positivity-preserving Well-balanced PAMPA Schemes with Global Flux quadrature for One-dimensional Shallow Water Models

本文提出一种保静水与非静水平衡的PAMPA方法，通过全局通量求积重构源项离散，并混合一阶局部Lax-Friedrichs格式以保水深正性，用于求解一维浅水模型。

Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocat…

一种用于周期表面散射的收敛数值方案：基于Floquet-Bloch变换

A Convergent Numerical Scheme for Scattering of Aperiodic Waves from Periodic Surfaces Based on the Floquet-Bloch Transform

本文提出一种基于Floquet-Bloch变换的收敛数值方案，用于解决非周期波对周期表面的散射问题，通过四次周期性单元的等效问题，避免了无界域的离散化需求，并通过正交近似和有限元方法进行收敛分析和误差估计。

牛顿-柯特斯公式的简单推导及实际误差

A Simple Derivation of Newton-Cotes Formulas with Realistic Errors

本文通过未定系数法和牛顿多项式基，推导出基于等距节点的插值型求积规则，并引入实际误差修正项，为经典牛顿-柯特斯公式提供了一种计算替代方案。

一种高效的高阶Nyström方案用于不均匀穿透介质的声散射

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

本文提出一种快速收敛的高阶Nyström方法，用于求解描述时谐声波在不均匀障碍物中散射的Lippmann-Schwinger积分方程，通过重叠坐标图表描述散射体，并利用分部单位简化高阶求积，解析处理积分算子中的奇异性，实现高阶精度。