Wasserstein interpolation with constraints and application to a parking problem

Abstract

We consider optimal transport problems where the cost for transporting a given probability measure µ 0 to another one µ 1 consists of two parts: the first one measures the transportation from µ 0 to an intermediate (pivot) measure µ to be determined (and subject to various constraints), and the second one measures the transportation from µ to µ 1 . This leads to Wasserstein interpolation problems under constraints for which we establish various properties of the optimal pivot measures µ. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Wasserstein constrained interpolation problems.

Entropic-Wasserstein Barycenters: PDE Characterization, Regularity, and CLT

“…For more general situations, we present some general tools to study Wasserstein medians, such as multi-marginal and dual formulations for the initial convex minimization problem. To the best of our knowldege, Wasserstein medians have not been very much investigated even in the Euclidean setting with more than two sample measures, however related optimal matching problems (with two sample measures and additional constraints) have been studied in [38], [31] and [15]. In the Euclidean setting in several dimensions, we also characterize medians by a minimal flow problem à la Beckmann [7] and a system of PDEs of Monge-Kantorovich type.…”