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Markov chains that describe interacting subsystems suffer from state space explosion but lead to highly structured matrices. In this work, we propose a novel tensor-based algorithm to address such tensor-structured Markov chains. Our algorithm combines a tensorized multigrid method with AMEn, an optimization-based low-rank tensor solver, for addressing coarse grid problems. Numerical experiments demonstrate that this combination overcomes the limitations incurred when using each of the twodoi:10.1553/etna_vol48s348 fatcat:hwpj4q6v4jdsjb66fo72qbdmpu