一种快速且谱收敛的有理阶分数积分和微分方程求解算法

Root-max Problems, Hybrid Expansion-Contraction, and Quadratically Convergent Optimization of Passive Systems

We present quadratically convergent algorithms to compute the extremal value of a real parameter for which a given rational transfer function of a linear time-invariant system is passive. This problem is formulated for both continuous-time and discrete-time systems and is linked to the problem of finding a realization of a rational transfer function such that its passivity radius is maximized. Our new methods make use of the Hybrid Expansion-Contraction algorithm, which we extend and generalize to the setting of what we call root-max problems.

迭代有理Krylov算法的收敛性

Convergence of the Iterative Rational Krylov Algorithm

本文针对状态空间对称系统，证明了迭代有理Krylov算法（IRKA）作为不动点迭代局部收敛于H2近似问题的局部极小值。

概率单纯形上复制器动力学的保结构有理积分器

A Structure-Preserving Rational Integrator for the Replicator Dynamics on the Probability Simplex

本文提出一种专为复制器动力学设计的二次收敛保结构有理积分器，通过两阶段有理近似与归一化步骤确保状态限制在概率单纯形并保持非负性、不变集和平衡点，理论分析与数值实验均验证了其二阶精度及优于传统求解器的性能。

Spectral mesh-free quadrature for planar regions bounded by rational parametric curves

This work presents spectral, mesh-free, Green's theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region's boundary curves corresponding to Green's theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green's theorem line integral. We present two methods to compute the intermediate quadrature r…

Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Here we study theoretically and compare experimentally an efficient method for solving systems of algebraic equations, where the matrix comes from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in $\mathbb R^d$, $d=1,2,3$. The proposed methods are based on the best uniform rational approximation (BURA) $r_{α,k}(t)$ of $t^α$ on $[0,1]$. Here $r_{α,k}$ is a rational function of $t$ involving numerator and denominator polynomials of degree at most $k$. …

共振评估：随机标量化边界积分预解式的自适应性与AAA有理逼近

Evaluation of resonances: adaptivity and AAA rational approximation of randomly scalarized boundary integral resolvents

提出一种基于随机标量化边界积分预解式的有理逼近、自适应搜索策略和指数收敛正割法终止阶段的新算法，用于评估开放和封闭腔体中的声学与电磁共振。