  The primal-dual method of Chambolle and Pock is a widely used algorithm to
solve various optimization problems written as convex-concave saddle point
problems. Each update step involves the application of both the forward linear
operator and its adjoint. However, in practical applications like computerized
tomography, it is often computationally favourable to replace the adjoint
operator by a computationally more efficient approximation. This leads to an
adjoint mismatch in the algorithm.
  In this paper, we analyze the convergence of Chambolle-Pock's primal-dual
method under the presence of a mismatched adjoint in the strongly convex
setting. We present an upper bound on the error of the primal solution and
derive stepsizes and mild conditions under which convergence to a fixed point
is still guaranteed. Furthermore we show linear convergence similar to the
result of Chambolle-Pock's primal-dual method without the adjoint mismatch.
Moreover, we illustrate our results both for an academic and a real-world
inspired application.

Optimization and Control;Numerical Analysis;Numerical Analysis