  When an inverse problem is solved by a gradient-based optimization algorithm,
the corresponding forward and adjoint problems, which are introduced to compute
the gradient, can be also solved iteratively. The idea of iterating at the same
time on the inverse problem unknown and on the forward and adjoint problem
solutions yields the concept of one-shot inversion methods. We are especially
interested in the case where the inner iterations for the direct and adjoint
problems are incomplete, that is, stopped before achieving a high accuracy on
their solutions. Here, we focus on general linear inverse problems and generic
fixed-point iterations for the associated forward problem. We analyze variants
of the so-called multi-step one-shot methods, in particular semi-implicit
schemes with a regularization parameter. We establish sufficient conditions on
the descent step for convergence, by studying the eigenvalues of the block
matrix of the coupled iterations. Several numerical experiments are provided to
illustrate the convergence of these methods in comparison with the classical
gradient descent, where the forward and adjoint problems are solved exactly by
a direct solver instead. We observe that very few inner iterations are enough
to guarantee good convergence of the inversion algorithm, even in the presence
of noisy data.
