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.
翻译:Kaczmarz 方法是一种迭代数字方法,用于解决大型和稀疏的线性方程式矩形系统。Gearhart、Koshy和Tam为Kaczmarz方法开发了加速技术,该技术最大限度地缩小了与全Kaczmarz步骤方向所需解决方案的距离。本文件将这一技术概括为加速方案,该技术将欧洲立方体规范差错降到最低程度,该方法由一些先前的迭代和卡茨马尔兹方法的另外一个周期组成。关键的挑战是如何找到一种公式,在其中了解确定独特最小最小最小度的最小度问题的所有参数,并有效解决这一问题。一个数字实验表明,拟议的扇形搜索有可能明显超越卡茨马尔兹和随机化的卡兹马尔兹方法,而不必进行Gearhart-Koshy/Tam线研究。