We provide rigorous theoretical bounds for Anderson acceleration (AA) that
allow for efficient approximate calculations of the residual, which reduce
computational time and memory storage while maintaining convergence.
Specifically, we propose a reduced variant of AA, which consists in projecting
the least squares to compute the Anderson mixing onto a subspace of reduced
dimension. The dimensionality of this subspace adapts dynamically at each
iteration as prescribed by computable heuristic quantities guided by the
theoretical error bounds. The use of the heuristic to monitor the error
introduced by approximate calculations, combined with the check on monotonicity
of the convergence, ensures the convergence of the numerical scheme within a
prescribed tolerance threshold on the residual. We numerically assess the
performance of AA with approximate calculations on: (i) linear deterministic
fixed-point iterations arising from the Richardson’s scheme to solve linear
systems with open-source benchmark matrices with various preconditioners and
(ii) non-linear deterministic fixed-point iterations arising from non-linear
time-dependent Boltzmann equations.