Fast Krasnosel’skiĭ–Mann Algorithm with a Convergence Rate of the Fixed Point Iteration of \(\boldsymbol{{ o} \left(\frac{1}{{ k}} \right)}\)

.The Krasnosel'skiĭ–Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skiĭ–Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of \(o \left(\frac{1}{k} \right)\) while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.Keywordsnonexpansive operatoraveraged operatorKrasnosel'skiĭ–Mann iterationNesterov's momentumLyapunov analysisconvergence ratesconvergence of iteratesMSC codes47J2047H0565K1565Y20