Although Anderson acceleration AA(m) has been widely used to speed up nonlinear solvers, most authors are simply using and studying the stationary version of Anderson acceleration. The behavior and full potential of the non-stationary version of Anderson acceleration methods remain an open question. Motivated by the hybrid linear solver GMRESR (GMRES Recursive), we recently proposed a set of non-stationary Anderson acceleration algorithms with dynamic window sizes AA(m,AA(n)) for solving both linear and nonlinear problems. Significant gains are observed for our proposed algorithms but these gains are not well understood. In the present work, we first consider the case of using AA(m,AA(1)) for accelerating linear fixed-point iteration and derive the polynomial residual update formulas for non-stationary AA(m,AA(1)). Like stationary AA(m), we find that AA(m,AA(1)) with general initial guesses is also a multi-Krylov method and possesses a memory effect. However, AA(m,AA(1)) has higher order degree of polynomials and a stronger memory effect than that of AA(m) at the k-th iteration, which might explain the better performance of AA(m,AA(1)) compared to AA(m) as observed in our numerical experiments. Moreover, we further study the influence of initial guess on the asymptotic convergence factor of AA(1, AA(1)). We show a scaling invariance property of the initial guess \(x_0\) for the AA(1,AA(1)) method in the linear case. Then, we study the root-linear asymptotic convergence factor under scaling of the initial guess and we explicitly indicate the dependence of root-linear asymptotic convergence factors on the initial guess. Lastly, we numerically examine the influence of the initial guess on the asymptotic convergence factor of AA(m) and AA(m,AA(n)) for both linear and nonlinear problems.

尽管安德森加速度 AA(m) 已广泛用于加速非线性求解器，但大多数作者只是简单地使用和研究安德森加速度的平稳版本。安德森加速方法的非平稳版本的行为和全部潜力仍然是一个悬而未决的问题。受混合线性求解器 GMRESR（GMRES 递归）的推动，我们最近提出了一组具有动态窗口大小 AA(m,AA(n)) 的非平稳安德森加速算法，用于求解线性和非线性问题。我们提出的算法观察到了显着的收益，但这些收益还没有得到很好的理解。在本工作中，我们首先考虑使用 AA(m,AA(1)) 加速线性定点迭代的情况，并推导非平稳 AA(m,AA(1)) 的多项式残差更新公式。与平稳 AA(m) 一样，我们发现具有一般初始猜测的 AA(m,AA(1)) 也是一种多 Krylov 方法，并且具有记忆效应。然而，AA(m,AA(1))在第k次迭代时比AA(m)具有更高的多项式阶数和更强的记忆效应，这可能解释了AA(m,AA(1)更好的性能)) 与我们的数值实验中观察到的 AA(m) 相比。此外，我们还进一步研究了初始猜测对AA(1, AA(1))渐近收敛因子的影响。我们展示了线性情况下 AA(1,AA(1)) 方法的初始猜测 \(x_0\) 的缩放不变性。然后，我们研究了初始猜测缩放下的根线性渐近收敛因子，并明确指出了根线性渐近收敛因子对初始猜测的依赖性。 最后，我们用数值方法检验了线性和非线性问题的初始猜测对 AA(m) 和 AA(m,AA(n)) 渐近收敛因子的影响。