Solution of Mismatched Monotone+Lipschitz Inclusion Problems

TL;DR

This paper develops and analyzes algorithms for solving monotone inclusion problems with mismatched adjoint operators, particularly in inverse problems like computed tomography, ensuring convergence despite operator approximations.

Contribution

It introduces variants of existing algorithms that handle mismatched Lipschitzian operators and proves their weak convergence under certain conditions.

Findings

Algorithms converge weakly despite operator mismatch

Applicable to inverse problems like computed tomography

Numerical experiments confirm theoretical results

Abstract

In this article, we study the convergence of algorithms for solving monotone inclusions in the presence of adjoint mismatch. The adjoint mismatch arises when the adjoint of a linear operator is replaced by an approximation, due to computational or physical issues. This occurs in inverse problems, particularly in computed tomography. In real Hilbert spaces, monotone inclusion problems involving a maximally $\rho$-monotone operator, a cocoercive operator, and a Lipschitzian operator can be solved by the Forward-Backward-Half-Forward and the Forward-Douglas-Rachford-Forward methods. We investigate the case of a mismatched Lipschitzian operator. We propose variants of the two aforementioned methods to cope with the mismatch, and establish conditions under which the weak convergence to a solution is guaranteed for these variants. The proposed algorithms hence enable each iteration to be implemented with a possibly iteration-dependent approximation to the mismatch operator, thus allowing this operator to be modified at each iteration. Finally, we present numerical experiments on a computed tomography example in material science, showing the applicability of our theoretical findings.