  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.
