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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.