High-Order Computation of Gradient Flows and Barycenters in Generalized Wasserstein Spaces

This dissertation presents a robust, scalable, and highly accurate numerical framework for computing gradient flows and barycenters in generalized Wasserstein spaces. These spaces are constructed by equipping the set of positive measures on a ground metric space with mean-field control-based extensions of the Benamou–Brenier formula. Gradient flow trajectories in such spaces represent solutions to a class of partial differential equations (PDEs) that model dissipative dynamics. A notable example is the porous medium equation which frequently appears in applications involving non-linear diffusion.

The first part of the work develops two classical first-order finite element schemes for solving the porous medium equation. These schemes are carefully designed to preserve key properties—mass conservation, positivity, and energy dissipation—implied by its underlying gradient flow structure. Their performance is validated through standard benchmark problems, and their convergence rates are numerically demonstrated.

Recognizing inherent limitations in extending classical methods to high-order temporal accuracy, a novel variational method is then introduced that obtains PDE solutions as gradient flow trajectories in Wasserstein spaces via mean-field control. This formulation is discretized using high-order spacetime finite elements and solved via the primal-dual hybrid gradient method. The efficacy of the numerical scheme is illustrated through application on a heat equation system and a porous medium equation system containing source terms, and high-order convergence rates are demonstrated in one and two spatial dimensions.

Building upon these developments, this work culminates in a generalized, mean-field control formulation of the Wasserstein barycenter problem, also discretized using high-order finite elements, and solved via the primal-dual hybrid gradient method.
The proposed formulation handles systems of unnormalized densities without relying on entropy regularization and incorporates reaction–diffusion dynamics. The robustness of the proposed method is demonstrated through several numerical experiments that compute barycenters in a three-dimensional Euclidean domain and on embedded two-dimensional manifolds.