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The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterativescheme designed to find a fixed point of an averaged operator in the frameworkof a real Hilbert space, since it lies at the heart of various numericalalgorithms for solving monotone inclusions and convex optimization problems. Weenhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates andshow that the resulting numerical method exhibits a convergence rate for thefixed point residual of $o(1/k)$ while preserving the weak convergence of theiterates to a fixed point of the operator. Numerical experiments illustrate thesuperiority of the resulting so-called Fast KM algorithm over various fixedpoint iterative schemes, and also its oscillatory behavior, which is a specificof Nesterov's momentum optimization algorithms.

Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o\left(\frac{1}{k}\right)$