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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 to 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.

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.; Dans ce travail nous nous intéressons à des problèmes inverses linéaires généraux où le problème direct correspondant est résolu de façon itérative en utilisant des méthodes de point fixe. Ainsi, les méthodes de type one-shot, qui itèrent en même temps sur la solution du problème direct et l’inconnue du problème inverse, peuvent être appliquées. Nous considérons deux variantes des méthodes multi-étapes de type one-shot et nous établissons des conditions suffisantes et nécessaires sur le pas de descente pour leur convergence, en étudiant les valeurs propres de la matrice par blocs des itérations couplées. Plusieurs tests numériques sont présentés pour illustrer la convergence de ces méthodes par rapport aux méthodes de descente de gradient usuelle et décentrée. En particulier, nous observons que très peu d’itérations internes pour le problème direct sont suffisantes pour garantir une bonne convergence de l’algorithme d’inversion.