In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel–Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel structure is constructed using a custom overlapping domain decomposition scheme with the notion of essential domains. As a solver for the constructed formulation, we propose a decoupled augmented Lagrangian method for untying the coupling of adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian method is provided. We present implementation details and numerical examples for various model problems including total variation minimizations and higher order models.

中文翻译：

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没有对偶成像问题的双重公式化的重叠域分解框架

在本文中，我们提出了一种新颖的重叠域分解方法，该方法可应用于变异成像中的各种问题，例如总变异最小化。最小化总变异最小的最新域分解方法大多数采用Fenchel-Rockafellar对偶性，而所提出的方法则基于原始公式。因此，所提出的方法不仅可以应用于总变化最小化，而且可以应用于那些具有复杂对偶问题的模型，例如高阶模型。在提出的方法中，使用自定义重叠域分解方案构造了具有并行结构的模型问题的等效公式，该方案具有基本域的概念。作为构造公式的求解器，我们提出了一种解耦的增强拉格朗日方法，以解开相邻子域的耦合。提供了解耦增强拉格朗日方法的收敛性分析。我们为各种模型问题（包括总变化最小化和高阶模型）提供了实现细节和数值示例。