Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis. (arXiv:2109.14176v2 [math.OC] UPDATED)

We study the asymptotic convergence of AA($m$), i.e., Anderson acceleration
with window size $m$ for accelerating fixed-point methods $x_{k+1}=q(x_{k})$,
$x_k \in R^n$. Convergence acceleration by AA($m$) has been widely observed but
is not well understood. We consider the case where the fixed-point iteration
function $q(x)$ is differentiable and the convergence of the fixed-point method
itself is root-linear. We identify numerically several conspicuous properties
of AA($m$) convergence: First, AA($m$) sequences $\{x_k\}$ converge
root-linearly but the root-linear convergence factor depends strongly on …

analysis arxiv convergence fixed-point linear math