Quantitative Stability and Numerical Resolution of the Moment Measure Problem

Abstract

The moment measure problem consists in finding a convex function $\ddot{I}ˆ$ whose moment measure, i.e., the pushforward by $\nabla \ddot{I}ˆ$ of the measure with density $e^{-\ddot{I}ˆ(\cdot )}$, is prescribed. It is highly non-linear and less understood than the related optimal transport problem. We establish a quantitative stability estimate for this problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.