This paper provides the first proof that Anderson acceleration (AA) improves 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 has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, 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 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 indicate that AA increases the radius of convergence. 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.

中文翻译：

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证明安德森加速提高了线性收敛定点方法的收敛率（但不是在二次收敛的方法中）

本文首次证明安德森加速（AA）提高了一般定点迭代的收敛速度。数十年来，AA 一直被用于在许多应用中加速非线性求解器，但是仍然缺乏对改进收敛速度的严格数学证明。这里提出的分析的关键思想是基于在 Hilbert 空间设置中执行内部优化将连续迭代的差异与残差相关联，并将优化阶段的增益明确定义为改进步骤的比率非加速定点迭代。我们证明的主要结果是 AA 将定点迭代的收敛速度提高到一阶，每一步都增加一个因子。除了提高收敛速度，我们的结果表明 AA 增加了收敛半径。最后，我们的估计表明，虽然提高了线性收敛速度，但估计中出现了额外的二次项，这说明了为什么 AA 通常不会提高二次收敛定点迭代的收敛性。给出了几个数值试验的结果，说明了该理论。