Duality-based Dynamical Optimal Transport of Discrete Time Systems

TL;DR

This paper introduces a novel duality-based algorithm for discrete-time dynamical optimal transport that avoids PDE solving, simplifies computations, and is highly parallelizable, with applications to linear Gaussian systems.

Contribution

It develops a new duality-based approach and a first-order splitting algorithm for discrete-time dynamical optimal transport, eliminating the need for PDE solutions and enabling efficient parallel computation.

Findings

The algorithm avoids solving PDEs in discrete-time optimal transport.

It simplifies the optimization to a maximization problem at each grid point.

Validation through simulation demonstrates effectiveness.

Abstract

We study dynamical optimal transport of discrete time systems (dDOT) with Lagrangian cost. The problem is approached by combining optimal control and Kantorovich duality theory. Based on the derived solution, a first order splitting algorithm is proposed for numerical implementation. While solving partial differential equations is often required in the continuous time case, a salient feature of our algorithm is that it avoids equation solving entirely. Furthermore, it is typical to solve a convex optimization problem at each grid point in continuous time settings, the discrete case reduces this to a straightforward maximization. Additionally, the proposed algorithm is highly amenable to parallelization. For linear systems with Gaussian marginals, we provide a semi-definite programming formulation based on our theory. Finally, we validate the approach with a simulation example.