  We consider nonlinear convergence acceleration methods for fixed-point
iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear
GMRES (NGMRES), and Nesterov-type acceleration (corresponding to AA with window
size one). We focus on fixed-point methods that converge asymptotically
linearly with convergence factor $\rho<1$ and that solve an underlying fully
smooth and non-convex optimization problem. It is often observed that AA and
NGMRES substantially improve the asymptotic convergence behavior of the
fixed-point iteration, but this improvement has not been quantified
theoretically. We investigate this problem under simplified conditions. First,
we consider stationary versions of AA and NGMRES, and determine coefficients
that result in optimal asymptotic convergence factors, given knowledge of the
spectrum of $q'(x)$ at the fixed point $x^*$. This allows us to understand and
quantify the asymptotic convergence improvement that can be provided by
nonlinear convergence acceleration, viewing $x_{k+1}=q(x_k)$ as a nonlinear
preconditioner for AA and NGMRES. Second, for the case of infinite window size,
we consider linear asymptotic convergence bounds for GMRES applied to the
fixed-point iteration linearized about $x^*$. Since AA and NGMRES are
equivalent to GMRES in the linear case, one may expect the GMRES convergence
factors to be relevant for AA and NGMRES as $x_k \rightarrow x^*$. Our results
are illustrated numerically for a class of test problems from canonical tensor
decomposition, comparing steepest descent and alternating least squares (ALS)
as the fixed-point iterations that are accelerated by AA and NGMRES. Our
numerical tests show that both approaches allow us to estimate asymptotic
convergence speed for nonstationary AA and NGMRES with finite window size.
