  In this work we are interested in general linear inverse problems where the
corresponding forward problem is solved iteratively using fixed point methods.
Then one-shot methods, which iterate at the same time on the forward problem
solution and on the inverse problem unknown, can be applied. We analyze two
variants of the so-called multi-step one-shot methods and establish sufficient
conditions on the descent step for their 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 usual and shifted gradient descent. In
particular, we observe that very few inner iterations on the forward problem
are enough to guarantee good convergence of the inversion algorithm.
