In this paper, we propose an overlapping additive Schwarz method for total variation minimization based on a dual formulation. The $O(1/n)$-energy convergence of the proposed method is proven, where $n$ is the number of iterations. In addition, we introduce an interesting convergence property called pseudo-linear convergence of the proposed method; the energy of the proposed method decreases as fast as linearly convergent algorithms until it reaches a particular value. It is shown that such the particular value depends on the overlapping width $\delta$, and the proposed method becomes as efficient as linearly convergent algorithms if $\delta$ is large. As the latest domain decomposition methods for total variation minimization are sublinearly convergent, the proposed method outperforms them in the sense of the energy decay. Numerical experiments which support our theoretical results are provided.

翻译：在本文中,我们提出了一个基于双重配方的重叠添加添加式施瓦兹方法,以全面减少变异。所提议方法的“O(1/n)$-能源融合”已被证明,其中美元为迭代数。此外,我们引入了一种有趣的趋同属性,称为“拟议方法的假线性趋同”;拟议方法的能量随着线性趋同算法的能量的下降而迅速下降,直到达到一个特定值。已经表明,这种特定值取决于重叠宽度$\delta$,而如果$\delta$是大,拟议方法的效率与线性趋同算法一样,如果$\delta$是大的话。由于全部变异最小化的最新域分解法是亚线性趋同,因此拟议的方法在能量衰减意义上优于它们。提供了支持我们理论结果的数值实验。