The Bregman-Kaczmarz method is an iterative method which can solve strongly convex problems with linear constraints and uses only one or a selected number of rows of the system matrix in each iteration, thereby making it amenable for large-scale systems. To speed up convergence, we investigate acceleration by heavy ball momentum in the so-called dual update. Heavy ball acceleration of the Kaczmarz method with constant parameters has turned out to be difficult to analyze, in particular no accelerated convergence for the L2-error of the iterates has been proven to the best of our knowledge. Here we propose a way to adaptively choose the momentum parameter by a minimal-error principle similar to a recently proposed method for the standard randomized Kaczmarz method. The momentum parameter can be chosen to exactly minimize the error in the next iterate or to minimize a relaxed version of the minimal error principle. The former choice leads to a theoretically optimal step while the latter is cheaper to compute. We prove improved convergence results compared to the non-accelerated method. Numerical experiments show that the proposed methods can accelerate convergence in practice, also for matrices which arise from applications such as computational tomography.

翻译：Bregman-Kaczmarz方法是一种迭代方法，可求解带线性约束的强凸问题，每次迭代仅使用系统矩阵的一行或选定行，因此适用于大规模系统。为加速收敛，我们研究了在所谓对偶更新中采用重球动量的加速策略。具有恒定参数的重球加速Kaczmarz方法难以分析，尤其据我们所知，尚未证明迭代L2误差具有加速收敛性。本文提出一种基于最小误差原理自适应选择动量参数的方法，该原理类似于近期针对标准随机Kaczmarz方法提出的方案。动量参数可通过精确最小化下一步迭代误差或最小化最小误差原理的松弛版本来选取：前者可达到理论最优步长，后者计算成本更低。我们证明了该方法相比非加速方法具有更优的收敛性。数值实验表明，所提方法在实际中可加速收敛，尤其适用于计算断层成像等应用场景生成的矩阵。