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This paper presents a general solution scheme for the problem of offline estimation of dynamic origin-destination (OD) demand matrices using traffic counts on some of the network links and historical demand information. The proposed method uses linear approximations of the assignment matrix, which maps the OD demand to link traffic counts. Several iterative algorithms that are based on this scheme are developed. The various algorithms are implemented in a tool that uses the mesoscopic traffic simulation model Mezzo to conduct network loadings. A case study network in Stockholm, Sweden, is used to test the proposed algorithms and to compare their performance with current state-of-the-art methods. The results demonstrate the applicability of the proposed methodology to efficiently obtain dynamic OD demand estimates for large and complex networks and that, computationally, this methodology outperforms existing methods.