Anderson acceleration (AA) is widely used for accelerating the convergence of nonlinear fixed-point methods $x_{k+1}=q(x_{k})$, $x_k \in \mathbb{R}^n$, but little is known about how to quantify the convergence acceleration provided by AA. As a roadway towards gaining more understanding of convergence acceleration by AA, we study AA($m$), i.e., Anderson acceleration with finite window size $m$, applied to the case of linear fixed-point iterations $x_{k+1}=M x_{k}+b$. We write AA($m$) as a Krylov method with polynomial residual update formulas, and derive recurrence relations for the AA($m$) polynomials. Writing AA($m$) as a Krylov method immediately implies that $k$ iterations of AA($m$) cannot produce a smaller residual than $k$ iterations of GMRES without restart (but without implying anything about the relative convergence speed of (windowed) AA($m$) versus restarted GMRES($m$)). We find that the AA($m$) residual polynomials observe a periodic memory effect where increasing powers of the error iteration matrix $M$ act on the initial residual as the iteration number increases. We derive several further results based on these polynomial residual update formulas, including orthogonality relations, a lower bound on the AA(1) acceleration coefficient $\beta_k$, and explicit nonlinear recursions for the AA(1) residuals and residual polynomials that do not include the acceleration coefficient $\beta_k$. Using these recurrence relations we also derive new residual convergence bounds for AA(1) in the linear case, demonstrating how the per-iteration residual reduction $||r_{k+1}||/||r_{k}||$ depends strongly on the residual reduction in the previous iteration and on the angle between the prior residual vectors $r_k$ and $r_{k-1}$. We apply these results to study the influence of the initial guess on the asymptotic convergence factor of AA(1), and to study AA(1) residual convergence patterns.

翻译：安德森加速(AA)被广泛用于加速非线性不动点迭代$x_{k+1}=q(x_{k})$（其中$x_k \in \mathbb{R}^n$）的收敛，但关于如何量化AA所提供的收敛加速知之甚少。为增进对AA收敛加速的理解，我们研究AA($m$)（即有限窗口大小$m$的安德森加速）在线性不动点迭代$x_{k+1}=M x_{k}+b$中的应用。我们将AA($m$)表述为一种具有多项式残差更新公式的Krylov方法，并推导了AA($m$)多项式的递推关系。将AA($m$)表述为Krylov方法直接表明：$k$次AA($m$)迭代产生的残差不会小于$k$次无重启GMRES迭代产生的残差（但这不涉及（带窗口的）AA($m$)与重启GMRES($m$)相对收敛速度的任何结论）。我们发现AA($m$)残差多项式具有周期性记忆效应，即随着迭代次数增加，误差迭代矩阵$M$的递增幂作用于初始残差。基于这些多项式残差更新公式，我们进一步推导了若干结果，包括正交关系、AA(1)加速系数$\beta_k$的下界，以及不包含加速系数$\beta_k$的AA(1)残差和残差多项式的显式非线性递推关系。利用这些递推关系，我们还为线性情况下的AA(1)推导了新的残差收敛界，展示了每步残差缩减量$||r_{k+1}||/||r_{k}||$如何强烈依赖于上一步的残差缩减量以及先前残差向量$r_k$与$r_{k-1}$之间的夹角。我们将这些结果用于研究初始猜测对AA(1)渐近收敛因子的影响，以及分析AA(1)的残差收敛模式。