In order to solve the homogenization problem, many FFT-based algorithms have been developed over the last decades. The AEM scheme ("Adaptive Eyre-Milton") has been recently introduced by Sab et al. , and is obtained by adapting Eyre-Milton’s one. It can efficiently handle the case of infinitely double-contrasted composites containing both pores and infinitely rigid inclusions. The authors have proved under some regularity assumptions that the AEM scheme is convergent for any choice of the reference material C0. Their numerical simulations have also shown that the optimal choice is to take C0 in the order of magnitude of the homogenized stiffness tensor Chom. For this choice, the convergence of the AEM scheme is faster than the other existing schemes. But if we choose a reference material that is very different from the homogenized one (several order of magnitudes), then the speed of convergence of the AEM scheme decreases. In this paper, we introduce a new version of the AEM scheme, where the reference ma- terial is updated iteratively, resulting in a very fast and versatile scheme which we call ’Accelerated AEM’ (A2EM). It will be shown that the accelerated algorithm is much faster than the regular one in the cases where C0 is several order of magnitude of Chom, while both schemes are comparable when C0 and Chom are identical.