A time-fractional optimal transport and mean-field planning: Formulation and algorithm

TL;DR

This paper introduces a novel framework for time-fractional optimal transport and mean-field planning, deriving coupled equations and proposing an efficient primal-dual algorithm, with numerical experiments demonstrating its effectiveness.

Contribution

It formulates a new time-fractional OT and MFP model, derives a coupled nonlinear system, and develops a preconditioned primal-dual algorithm for efficient numerical solution.

Findings

Effective algorithm accelerates convergence for fractional PDEs.

Numerical experiments validate the model's ability to handle Gaussian and image densities.

Proposed method demonstrates flexibility and robustness in various scenarios.

Abstract

The time-fractional optimal transport (OT) and mean-field planning (MFP) models are developed to describe the anomalous transport of the agents in a heterogeneous environment such that their densities are transported from the initial density distribution to the terminal one with the minimal cost. We derive a strongly coupled nonlinear system of a time-fractional transport equation and a backward time-fractional Hamilton-Jacobi equation based on the first-order optimality condition. The general-proximal primal-dual hybrid gradient (G-prox PDHG) algorithm is applied to discretize the OT and MFP formulations, in which a preconditioner induced by the numerical approximation to the time-fractional PDE is derived to accelerate the convergence of the algorithm for both problems. Numerical experiments for OT and MFP problems between Gaussian distributions and between image densities are carried out to investigate the performance of the OT and MFP formulations. Those numerical experiments also demonstrate the effectiveness and flexibility of our proposed algorithm.