Alternating Direction Method of Multipliers for Linear Inverse Problems

TL;DR

This paper introduces an ADMM-based iterative method for solving linear inverse problems in Hilbert spaces with convex penalties, providing convergence analysis and demonstrating its effectiveness as a regularization technique through numerical tests.

Contribution

The paper develops a novel ADMM algorithm for linear inverse problems in Hilbert spaces, including convergence analysis without Lagrange multipliers and regularization properties with noisy data.

Findings

Convergence of the ADMM algorithm without Lagrange multipliers.

The method acts as a regularization technique with noisy data.

Numerical simulations confirm the efficiency of the proposed method.

Abstract

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of Lagrange multiplier. In case the data contains noise, we show that our method is a regularization method as long as it is terminated by a suitable stopping rule. Various numerical simulations are performed to test the efficiency of the method.