The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas–Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered–reflection method is able to both accelerate the average reflection scheme by the Douglas–Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of alternating projections. Linear convergence for this novel block-wise circumcenter framework is derived and illustrated numerically. Furthermore, we prove that the original circumcentered–reflection method essentially finds the best approximation solution in one single step if the given affine subspaces are hyperplanes.

中文翻译：

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块状外心反射法

最近已使用基本欧几里得中心点概念来改进经典Douglas-Rachford方法的两个方面，以投射到仿射子空间的交点上。所谓的“圆心反射”方法既可以加速道格拉斯-拉奇福德方法的平均反射方案，又可以处理两个以上仿射子空间的交集。现在，我们介​​绍在块中进行外接圆心化的技术，这不仅仅是对外接圆心的基本算法的一种选择，事实证明，这是概括交替投影方法的一种优雅方式。得出了这种新颖的块状外接中心框架的线性收敛性，并用数值进行了说明。此外，