  The state-of-art seismic imaging techniques treat inversion tasks such as FWI
and LSRTM as PDE-constrained optimization problems. Due to the large-scale
nature, gradient-based optimization algorithms are preferred in practice to
update the model iteratively. Higher-order methods converge in fewer iterations
but often require higher computational costs, more line search steps, and
bigger memory storage. A balance among these aspects has to be considered. We
propose using Anderson acceleration (AA), a popular strategy to speed up the
convergence of fixed-point iterations, to accelerate the steepest descent
algorithm, which we innovatively treat as a fixed-point iteration. Independent
of the dimensionality of the unknown parameters, the computational cost of
implementing the method can be reduced to an extremely low-dimensional
least-squares problem. The cost can be further reduced by a low-rank update. We
discuss the theoretical connections and the differences between AA and other
well-known optimization methods such as L-BFGS and the restarted GMRES and
compare their computational cost and memory demand. Numerical examples of FWI
and LSRTM applied to the Marmousi benchmark demonstrate the acceleration
effects of AA. Compared with the steepest descent method, AA can achieve fast
convergence and provide competitive results with some quasi-Newton methods,
making it an attractive optimization strategy for seismic inversion.
