Most optimal transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. Another route is to approximate moments of measures, when these are determinate. Here, we show that many common OT problems can be formulated as generalized moment problems [1]. These include the computation of Lp Wasserstein or Gromov-Wasserstein distances and the corresponding barycenters. For measures supported on compact semi-algebraic sets, a practical numerical approach then consists in truncating moment sequences up to a certain order, and using the polynomial sums-of-squares hierarchy. The approach is proved to converge. It allows to extract a posteriori relevant information on the measures, such as their supports that can be estimated using Christoffel-Darboux kernels. In a second part of this talk, we present how these moment methods can be used for the solution of parameter-dependent hyperbolic equations, by using weak formulations on measure-valued solutions [2]. These methods can be used to directly approximate the solution as a function of the parameter, or to construct reduced order models by interpolation in Wasserstein spaces.
These are joint works with Olga Mula and Clément Cardoen, Swann Marx and Nicolas Seguin.
References:
[1] O. Mula and A. Nouy. Moment-SoS methods for optimal transport problems, arXiv:2211.10742, 2022.
[2] C. Cardoen, S. Marx, A. Nouy, and N. Seguin. A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws, 2024.