Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper, we tackle the challenge of deriving convergence rates for Landweber-type methods applied to ill-posed inverse problems, where forward operators map from a Banach space to a Hilbert space. Under a benchmark source condition, we introduce a novel strategy to derive convergence rates when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. Our results offer substantial flexibility regarding step sizes, by allowing the use of variable step sizes. By extending the strategy to deal with the stochastic mirror descent method for solving nonlinear ill-posed systems with exact data, under a benchmark source condition we also obtain an almost sure convergence rate in terms of the number of iterations.

翻译：Landweber型方法是求解Banach空间中不适定反问题的重要方法，其收敛性已得到充分理解。然而，如何推导其收敛速率仍是一个具有挑战性的开放性问题。本文致力于解决Landweber型方法应用于不适定反问题时的收敛速率推导难题，其中前向算子从Banach空间映射到Hilbert空间。在基准源条件下，我们提出了一种新颖的策略，用于推导该方法在采用先验停止准则或偏差原则终止时的收敛速率。我们的结果在步长选择上具有很大灵活性，允许使用可变步长。通过将该策略扩展至处理具有精确数据的非线性不适定系统的随机镜像下降方法，在基准源条件下我们还获得了关于迭代次数的几乎必然收敛速率。