In this work, we aim at efficiently solving a parametrized family of optimal transport problems by using model order reduction methods. We propose a reduced-order model by adding to the primal (respectively dual) version of the high-fidelity model the additional constraint to live in a non negative sub cone (resp. in subspaces) of small dimension. The reduced-order model then reads as a linear program with a small number of degrees of freedom and constraints. We identify explicit conditions under which this reduced-order model has at least one solution. We propose two a posteriori error estimations that bounds the error between the optimal values of the high-fidelity problem and the reduced-order model. As one of these estimations requires the computation of non linear terms (with respect to the reduction of dimension), we use an Empirical Interpolation Method (EIM) (see e. g. maday2007general or barrault2004empirical) to numerically efficiently compute this estimation. We apply the whole methodology on a simple 1D example and on a problem of color transfer between images, and compare its performances to Sinkhorn algorithm.
Bonnet-Weill et al. (Fri,) studied this question.