  A simple procedure to map two probability measures in $\mathbb{R}^d$ is the
so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging
monotonically the marginal distributions of the last coordinate, and then the
conditional distributions, iteratively. We show that this mapping is the limit
of solutions to a class of Monge-Kantorovich mass transportation problems with
quadratic costs, with the weights of the coordinates asymptotically dominating
one another. This enables us to design a continuation method for numerically
solving the optimal transport problem.
