[Paper Review] An algorithm for solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators

Radu Ioan Boţ, Christopher Hendrich|arXiv (Cornell University)|Jun 13, 2013

Numerical methods in inverse problems5 citations

TL;DR

This paper proposes two primal-dual algorithms for solving structured monotone inclusion problems involving parallel sums of linearly composed maximally monotone operators, leveraging forward-backward splitting techniques to handle each operator individually. The key contribution lies in enabling efficient solution of imaging problems with first- and second-order total variation functionals through specialized treatment of parallel sums.

ABSTRACT

The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some elaborated splitting techniques, all of the operators occurring in the problem formulation are processed individually via forward or backward steps. The treatment of parallel sums of linearly composed maximally monotone operators is motivated by applications in imaging which involve first- and second-order total variation functionals, to which a special attention is given.

Motivation & Objective

• To address structured monotone inclusion problems involving parallel sums of compositions of maximally monotone operators and linear bounded operators.
• To develop efficient numerical methods that process each operator individually using forward or backward steps.
• To enable practical solution of imaging problems involving first- and second-order total variation functionals through tailored algorithmic treatment.
• To extend existing splitting methods to handle parallel sums in a way that preserves convergence and computational efficiency.

Proposed method

• The algorithms employ a primal-dual forward-backward splitting framework to handle individual operators in the inclusion problem.
• Parallel sums of linearly composed maximally monotone operators are treated via specialized splitting techniques that decouple the components.
• Forward steps are applied to monotone operators with Lipschitz continuous gradients, while backward steps are used for maximal monotonicity.
• The method explicitly handles compositions of maximally monotone operators with bounded linear operators using operator splitting strategies.
• The algorithm design ensures that all operators are processed independently, enhancing modularity and computational tractability.
• The approach is validated through applications to imaging problems with total variation regularization, particularly those involving first- and second-order variations.

Experimental results

Research questions

• RQ1 How can structured monotone inclusion problems involving parallel sums of linearly composed maximally monotone operators be solved efficiently?
• RQ2 What splitting techniques enable individual processing of each operator in the inclusion via forward or backward steps?
• RQ3 How can the proposed algorithms be adapted to imaging problems with first- and second-order total variation functionals?
• RQ4 What theoretical convergence guarantees can be established for the primal-dual methods under the given problem structure?

Key findings

• The proposed algorithms achieve convergence for structured monotone inclusions involving parallel sums of linearly composed maximally monotone operators.
• The treatment of parallel sums enables effective handling of imaging problems with first- and second-order total variation functionals.
• Individual operator processing via forward and backward steps ensures computational efficiency and modularity.
• The method supports applications in imaging where total variation regularization is used, particularly in problems involving higher-order smoothness constraints.
• The algorithmic framework is flexible and extensible to various compositions of monotone operators and linear operators.
• The approach provides a practical and theoretically grounded solution for complex monotone inclusion problems arising in imaging and optimization.