JKO schemes with general transport costs

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.