Generalized Fast Krasnosel'kiĭ-Mann Method with Preconditioners

We study accelerated Krasnosel'ki\u{\i}-Mann-type methods with preconditioners in both continuous and discrete time. From a continuous time model, we derive a generalized fast Krasnosel'ki\u{\i}-Mann method, providing a new yet simple proof of convergence that allows for unprecedented flexibility in parameter tuning. Our analysis unifies inertial and anchoring acceleration mechanisms and offers a broad range of parameter choices, which prove beneficial in practice. Additionally, we extend our analysis to the case where preconditioners are allowed to be degenerate, unifying the treatment of various splitting methods and establishing the weak convergence of shadow sequences.