Context and background
Wasserstein distances are a powerful tool to compare data in a geometrically meaningful way that does not rely on a specific vector representation. While significant progress has been made in using these distances for data interpolation, many applications – such as statistical analysis of measured valued data, reduced order modeling, or the construction of numerical methods for physics simulations – require not only interpolation but also extrapolation of data. However, there is currently only a partial understanding of how to define such extrapolations consistently with the Wasserstein geometry. One promising direction is based on the notion of Wasserstein barycenter with signed weights, which has been recently investigated in [1] in a simplified setting.
Objectives
The main objective of this thesis is to expand our understanding of Wasserstein barycenters, focussing in particular on the case with signed weights. In particular, we aim to produce a computationally tractable characterization of such objects and provide efficient numerical algorithms for their computation. We will also invesitigate different applications of the developed tools, including:
- the discretization of Wasserstein gradient flows, developing the ideas of [2];
- the discretization of fluid models such as the pressureless and isentropic Euler equations;
- the development of accelerated schemes for optimization on the space of measures.
[1] Thomas O. Gallouët, Andrea Natale, and Gabriele Todeschi. Metric extrapolation in the wasserstein space, 2025. Calculus of Variations and Partial Differential Equations
[2] Thomas O. Gallouët, Andrea Natale, and Gabriele Todeschi. From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows. Mathematics of Computation, 93:2769–2810, 2024.
