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Related Experiment Video

Updated: May 5, 2026

Models and Methods to Evaluate Transport of Drug Delivery Systems Across Cellular Barriers

Models and Methods to Evaluate Transport of Drug Delivery Systems Across Cellular Barriers

Published on: October 18, 2013

JKO schemes with general transport costs.

Cale Rankin¹, Ting-Kam Leonard Wong²

¹Department of Mathematics, Monash University, Victoria, Australia.

Calculus of Variations and Partial Differential Equations

|February 20, 2026

Summary

We modified the JKO scheme for Wasserstein gradient flow by using general transport costs on manifolds. This modified scheme converges to the Riemannian Fokker-Planck equation, offering computational advantages.

Keywords:

35K57 (Primary)58J35 82C31 (Secondary)

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Area of Science:

Numerical Analysis
Differential Geometry
Stochastic Processes

Background:

The JKO scheme is a time discretization for Wasserstein gradient flows.
Wasserstein distance can be computationally intensive on manifolds.
Fokker-Planck equations model diffusion processes.

Purpose of the Study:

To generalize the JKO scheme using arbitrary transport costs on manifolds.
To establish convergence to the Riemannian Fokker-Planck equation.
To explore computational alternatives to the Riemannian distance.

Main Methods:

Modification of the JKO scheme by replacing Wasserstein distance with general transport costs.
Analysis of convergence properties under conditions on the cost function's Hessian.
Application to Fokker-Planck equations on compact and complete Riemannian manifolds.

Main Results:

Convergence of the modified JKO scheme to the Riemannian Fokker-Planck equation when the cost induces a Riemannian metric.
Demonstration of applicability on compact submanifolds with Neumann boundary conditions and complete Riemannian manifolds.
Successful application to Hessian manifolds using Bregman divergence as a cost.

Conclusions:

The generalized JKO scheme provides a flexible framework for discretizing gradient flows on manifolds.
This approach offers computational benefits when Riemannian distance is intractable.
The method connects optimal transport, geometric analysis, and numerical methods effectively.