Convergence Results for Primal-Dual Algorithms in the Presence of Adjoint Mismatch

Most optimization problems arising in imaging science involve high-dimensional linear operators and their adjoints. In the implementations of these operators, changes may be introduced for various practical considerations (e.g., memory limitation, computational cost, convergence speed), leading to an adjoint mismatch. This occurs for the X-ray tomographic inverse problems found in computed tomography (CT), where a surrogate operator often replaces the adjoint of the measurement operator (called the projector). The resulting adjoint mismatch can jeopardize the convergence properties of iterative schemes used for image recovery. In this paper, we study the theoretical behavior of a panel of primal-dual proximal algorithms, which rely on forward-backward-(forward) splitting schemes when an adjoint mismatch occurs. We analyze these algorithms by focusing on the resolution of possibly nonsmooth convex penalized minimization problems in an infinite-dimensional setting. Using tools from fixed point theory, we show that they can solve monotone inclusions beyond minimization problems. Such findings indicate that these algorithms can be seen as a generalization of classical primal-dual formulations. The applicability of our findings is also demonstrated through two numerical experiments in the context of CT image reconstruction.