相关论文: Primal dual methods for Wasserstein gradient flows

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the…

We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a…

Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are…

We propose a Variable-Preconditioned Transformed Primal-Dual (VPTPD) method for solving generalized Wasserstein gradient flows based on the structure-preserving JKO scheme. This is a nontrivial extension of the TPD method [Chen et al.…

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main…

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to…

We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our…

We study the distributionally robust optimization (DRO) in a dynamic context where the model uncertainty is captured by penalizing potential models in function of their adapted Wasserstein distance to a given reference model. We consider…

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures…

The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamics formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE…

Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss…

Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence…

In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model…

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in…

Gradient flows are a powerful tool for optimizing functionals in general metric spaces, including the space of probabilities endowed with the Wasserstein metric. A typical approach to solving this optimization problem relies on its…

We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows…

Over the past fifteen years, the theory of Wasserstein gradient flows of convex (or, more generally, semiconvex) energies has led to advances in several areas of partial differential equations and analysis. In this work, we extend the…

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of…

We study the discretization of generalized Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of…

This paper aims to investigate a full numerical approximation of non-autonomous semilnear parabolic partial differential equations (PDEs) with nonsmooth initial data. Our main interest is on such PDEs where the nonlinear part is stronger…