5 papers
A Unified Variational Framework for Optimal Transport with Lagrangian Costs
Hailiang Liu
We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a…
A Primal-Dual Level Set Method for Computing Geodesic Distances
Hailiang Liu, Laura Zinnel
The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance…
A Generalized Energy-Based Adaptive Gradient Method for Optimization
Lin Feng, Hailiang Liu
Adaptive Gradient Descent with Energy (AEGD) is a variant of Gradient Descent (GD) designed to address step size sensitivity through an energy-based formulation. AEGD is notable fo…
Variational structure of Fokker-Planck equations with variable mobility
Hailiang Liu, Athanasios E. Tzavaras
We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced…
The Onsager principle and structure preserving numerical schemes
Huangxin Chen, Hailiang Liu, Xianmin Xu
We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial d…