Accelerated Additive Schwarz Methods for Convex Optimization with Adaptive Restart

Abstract

Based on an observation that additive Schwarz methods for general convex optimization can be interpreted as gradient methods, we propose an acceleration scheme for additive Schwarz methods. Adopting acceleration techniques developed for gradient methods such as momentum and adaptive restarting, the convergence rate of additive Schwarz methods is greatly improved. The proposed acceleration scheme does not require any a priori information on the levels of smoothness and sharpness of a target energy functional, so that it can be applied to various convex optimization problems. Numerical results for linear elliptic problems, nonlinear elliptic problems, nonsmooth problems, and nonsharp problems are provided to highlight the superiority and the broad applicability of the proposed scheme.

20 pages, 8 figures

References in corpus (4)

• Additive Schwarz Methods for Convex Optimization as Gradient Methods
• Fast Nonoverlapping Block Jacobi Method for the Dual Rudin--Osher--Fatemi Model
• A Finite Element Nonoverlapping Domain Decomposition Method with Lagrange Multipliers for the Dual Total Variation Minimizations
• Pseudo-linear Convergence of an Additive Schwarz Method for Dual Total Variation Minimization

Cited by in corpus (3)

• Additive Schwarz Methods for Convex Optimization with Backtracking
• Additive Schwarz methods for fourth-order variational inequalities
• On the linear convergence of additive Schwarz methods for the $p$-Laplacian