  We provide rigorous theoretical bounds for Anderson acceleration (AA) that
allow for approximate calculations when applied to solve linear problems. We
show that, when the approximate calculations satisfy the provided error bounds,
the convergence of AA is maintained while the computational time could be
reduced. We also provide computable heuristic quantities, guided by the
theoretical error bounds, which can be used to automate the tuning of accuracy
while performing approximate calculations. For linear problems, the use of
heuristics to monitor the error introduced by approximate calculations,
combined with the check on monotonicity of the residual, ensures the
convergence of the numerical scheme within a prescribed residual tolerance.
Motivated by the theoretical studies, we propose a reduced variant of AA, which
consists in projecting the least-squares used to compute the Anderson mixing
onto a subspace of reduced dimension. The dimensionality of this subspace
adapts dynamically at each iteration as prescribed by the computable heuristic
quantities. We numerically show and 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.
