In this paper, we propose and analyze a set of fully non-stationary Anderson acceleration algorithms with dynamic window sizes and optimized damping. Although Anderson acceleration (AA) has been used for decades to speed up nonlinear solvers in many applications, most authors are simply using and analyzing the stationary version of Anderson acceleration (sAA) with fixed window size and a constant damping factor. The behavior and potential of the non-stationary version of Anderson acceleration methods remain an open question. Since most efficient linear solvers use composable algorithmic components. Similar ideas can be used for AA to solve nonlinear systems. Thus in the present work, to develop non-stationary Anderson acceleration algorithms, we first propose two systematic ways to dynamically alternate the window size $m$ by composition. One simple way to package sAA(m) with sAA(n) in each iteration is applying sAA(m) and sAA(n) separately and then average their results. It is an additive composite combination. The other more important way is the multiplicative composite combination, which means we apply sAA(m) in the outer loop and apply sAA(n) in the inner loop. By doing this, significant gains can be achieved. Secondly, to make AA to be a fully non-stationary algorithm, we need to combine these strategies with our recent work on the non-stationary Anderson acceleration algorithm with optimized damping (AAoptD), which is another important direction of producing non-stationary AA and nice performance gains have been observed. Moreover, we also investigate the rate of convergence of these non-stationary AA methods under suitable assumptions. Finally, our numerical results show that some of these proposed non-stationary Anderson acceleration algorithms converge faster than the stationary sAA method and they may significantly reduce the storage and time to find the solution in many cases.

翻译：在本文中,我们提出并分析一套完全非静止的安德森加速算法。 由于最有效的线性解解算器使用可变算法组件。 类似的想法可用于 AA 解析非线性安德森加速算法。 因此, 在目前的工作中, 开发非静止的安德森加速算法( AA AA ), 我们首先提出两种系统的方法, 以动态的方式将窗口加速算法( SA AA ) 的固定版本( SA AA ) 用于在许多应用程序中加速非线性解算法( SA A A ), 多数作者只是使用和分析固定窗口大小和常态阻断因素的固定版本。 非静止的安德森加速算法( SA ) 行为和非静止的加速算法( sA (n) 单独使用SA (m) 和 sA (n) 平均计算结果。 这是一种添加式的混合方法。 另一种更重要的方式是多复制性复合组合, 也就是说, 我们用这个高级的 A (A ) 最终将一个重要的直流性计算结果, 与另一个直径 A 。