Optimal transport theory has recently re-emerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology—the reconstruction of the linear density field from low redshifts, in particular the recovery of the baryonic acoustic oscillation (BAO) scale. We demonstrate our algorithm’s accuracy by retrieving the BAO scale in noiseless cosmological simulations that are dedicated to cancel cosmic variance; we find uncertainties to be reduced by a factor of 4.3 compared with performing no reconstruction, and a factor of 3.1 compared with standard reconstruction.