论文标题
kaczmarz方法的广义齿轮 - koshy加速度
Generalized Gearhart-Koshy acceleration for the Kaczmarz method
论文作者
论文摘要
kaczmarz方法是一种迭代数值方法，用于求解线性方程的大而稀疏的矩形系统。 Gearhart，Koshy和Tam已开发了一种用于Kaczmarz方法的加速技术，该技术可以最大程度地减少到完整的Kaczmarz步骤方向上与所需溶液的距离。 本文将该技术概括为加速方案，该方案将欧几里得规范误差最小化，而欧几里得型误差是由许多先前的迭代和kaczmarz方法的一个额外循环跨越的仿射子空间。关键的挑战是找到一种公式，其中最小二乘问题的所有参数均定义了独特的最小化器，并有效地解决了此问题。 数值实验表明，提出的仿射搜索有可能明确胜过Kaczmarz和带有和不带有Gearhart-Koshy/TAM线路搜索的随机Kaczmarz方法。
The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step. The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently. A numerical experiment demonstrates that the proposed affine search has the potential to clearly outperform the Kaczmarz and the randomized Kaczmarz methods with and without the Gearhart-Koshy/Tam line-search.
