Solution of Mismatched Monotone+Lipschitz Inclusion Problems

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.