Pseudo-linear Convergence of an Additive Schwarz Method for Dual Total Variation Minimization

TL;DR

This paper introduces an overlapping additive Schwarz method for dual total variation minimization, demonstrating $O(1/n)$ convergence and a pseudo-linear convergence property that enhances efficiency over existing sublinear methods.

Contribution

The paper presents a novel Schwarz method with proven $O(1/n)$ convergence and a unique pseudo-linear convergence behavior, improving upon existing sublinear domain decomposition approaches.

Findings

The method achieves $O(1/n)$ energy convergence.

It exhibits pseudo-linear convergence until reaching a certain energy level.

Numerical results confirm theoretical convergence properties.

Abstract

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.