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.

翻译：在这项工作中,我们感兴趣的是一般的线性反向问题,在这些问题中,相应的前方问题用固定点方法反复解决。然后,可以采用一发方法,同时对前方问题解决办法和未知的反面问题进行反复处理。我们分析了所谓的多步一发方法的两个变式,并通过研究相交迭交错的区块矩阵的精细值,在下降步骤上为其趋同创造充分的条件。提供了若干数字实验,以说明这些方法与典型的惯用和偏移梯度梯度下降方法的趋同。我们特别注意到,关于前方问题的内部迭代很少足以保证反向算法的良好趋同。