Optimal Transport: Theory, Numerical Methods and Applications in Applied Mathematics

Abstract

Optimal transport asks for the cheapest way of rearranging one distribution of mass into another, and the answer supplies a metric on probability measures that respects the geometry of the underlying space. This review traces the subject from the formulation posed by Monge in 1781, through the linear relaxation introduced by Kantorovich, to the characterisation of optimal maps as gradients of convex functions. The resulting Wasserstein distances behave differently from divergences based on pointwise comparison of densities: they remain finite and informative for measures with disjoint support, and the shortest paths they induce displace mass rather than fade one profile into another. The dynamic formulation connects these paths to a fluid mechanical problem and identifies several diffusion equations as gradient flows of familiar energies. Numerical work has changed the field. Entropic regularisation reduces the linear program to matrix scaling iterations that are simple, parallel and differentiable, at the cost of blurring the transport plan by an amount controlled by the regularisation parameter. Semi-discrete and sliced methods offer alternatives when accuracy matters more than speed. Applications now include image retrieval, generative modelling, domain adaptation and the reconstruction of developmental trajectories from single-cell measurements.