A proof that Anderson acceleration increases the convergence rate in linearly converging fixed point methods (but not in quadratically converging ones)

This paper provides the first proof that Anderson acceleration (AA) increases the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in many applications, however a rigorous mathematical justification of the improved convergence rate remained lacking. The key ideas of the proof are relating errors with residuals, using results arising from the optimization, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that AA improves a the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. In addition to improving the convergence rate, our results also show that AA increases the radius of convergence (even beyond a set where the fixed point operator is contractive). Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations. Results of several numerical tests are given which illustrate the theory.