State-of-the-art seismic imaging techniques treat inversion tasks such as full-waveform inversion (FWI) and least-squares reverse time migration (LSRTM) as partial differential equation-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 have conducted an evaluation 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 unknown parameter dimensionality, 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 determine the theoretical connections and the differences between AA and other well-known optimization methods such as L-BFGS and the restarted generalized minimal residual method and compare their computational cost and memory requirements. 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 faster convergence and can provide competitive results with some quasi-Newton methods, making it an attractive optimization strategy for seismic inversion.