On a progressive and iterative approximation method with memory for least square fitting

A Progressive Statistical Method for Preconditioning Matrix-Free Solution of High-Order Discretization of Linear Time-Dependent Problems

Preconditioning of a linear system obtained from spectral discretization of time-dependent PDEs often results in a full matrix which is expensive to compute and store specially when the problem size increases. A matrix-free implementation is usually applied to resolve this issue. In this framework, preconditioning is typically challenging since the entries of the matrix are not explicitly available. In this short note, we propose a statistical approach to gradually create a preconditioner matrix by collecting the information obtained from matrix-vector product in the Arnoldi loop of an unpreco…

三次B样条插值曲线的预处理几何迭代方法

Preconditioned geometric iterative methods for cubic B-spline interpolation curves

本文针对三次B样条插值曲线，提出预处理几何迭代方法，通过构造预处理器对渐进迭代逼近及其变体进行预处理，在可忽略额外计算成本下有效加速了收敛速率。

重加权最小二乘拟合的一般公式化表述

A general formulation of reweighted least squares fitting

本文提出重加权最小二乘逼近的广义公式，证明有限维空间中解可表示为插值函数的凸组合，并给出依据误差迭代更新权重的策略，通过多项式与样条数值实验验证其在曲线、曲面及自适应样条拟合中的效果。

Clothoid Fitting and Geometric Hermite Subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To define clothoidal averaging, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and fo…

将随机迭代求解器视为迭代细化：实现后向稳定性的简单修正

Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability

本文将随机迭代求解器视为迭代细化，提出结合迭代与递归细化的新算法SIRR，仅通过重组计算顺序即实现后向误差四个数量级改进，是首个兼具前向和后向稳定性的渐近快速单阶段随机最小二乘求解器。