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.

翻译：当通过基于梯度的优化算法求解反问题时，用于计算梯度的正演和伴随问题也可以通过迭代方式求解。同时迭代反问题未知量以及正演和伴随问题解的思想催生了"一步反演方法"的概念。我们特别关注正演和伴随问题的内迭代不完整的情形，即在其解达到高精度之前提前终止迭代。本文聚焦于一般线性反问题及其正演问题的通用不动点迭代，分析了所谓"多步一步方法"的变体，特别是带有正则化参数的半隐式格式。通过研究耦合迭代分块矩阵的特征值，我们建立了下降步长收敛的充分条件。为说明这些方法的收敛性，我们提供了多个数值实验，并将其与经典梯度下降法（该法通过直接求解器精确求解正演和伴随问题）进行对比。实验结果表明，即便在数据含有噪声的情况下，极少数内迭代次数即可保证反演算法具有良好的收敛性。