Efficient Primal-dual Forward-backward Splitting Method for Wasserstein-like Gradient Flows with General Nonlinear Mobilities
Yunhong Deng, Li Wang, Chaozhen Wei
TL;DR
This work develops a novel saddle-point reformulation for the JKO scheme of Wasserstein-like gradient flows with general nonlinear mobilities and solves it via a primal-dual forward-backward (PDFB) splitting method. The approach decouples mobility from the action using a Legendre–Fenchel transform, enabling a fully discrete, structure-preserving scheme on staggered grids that preserves energy dissipation, mass, and positivity. A rigorous convergence analysis shows ergodic convergence of the PDFB iterates with step-size conditions that can be made grid-size independent, and the method is accelerated by convex-splitting as nonlinear preconditioning. Numerical experiments in 1D and 2D, including porous media diffusion, saturated Fokker–Planck, thin-film lubrication, and Cahn–Hilliard-type models, demonstrate accurate, stable performance and computational efficiency, outperforming previous proximal-splitting approaches. The framework is readily adaptable to a broad class of nonlinear mobilities, offering a practical and scalable tool for simulating Wasserstein-like gradient flows.
Abstract
We construct an efficient primal-dual forward-backward (PDFB) splitting method for computing a class of minimizing movement schemes with nonlinear mobility transport distances, and apply it to computing Wasserstein-like gradient flows. This approach introduces a novel saddle point formulation for the minimizing movement schemes, leveraging a support function form from the Benamou-Brenier dynamical formulation of optimal transport. The resulting framework allows for flexible computation of Wasserstein-like gradient flows by solving the corresponding saddle point problem at the fully discrete level, and can be easily extended to handle general nonlinear mobilities. We also provide a detailed convergence analysis of the PDFB splitting method, along with practical remarks on its implementation and application. The effectiveness of the method is demonstrated through several challenging numerical examples.