Limit Theorems for Optimal Mass Transportation

最优物质运输与马瑟理论

Optimal mass transportation and Mather theory

我们研究的最优运输措施紧流形上的成本定义从凸拉格朗日。我们证明，最佳运输可以插测Lipschitz叠层，或几何电流。该方法的灵感来自于拉格朗日系统的马瑟理论。我们利用黏性解的相关Hamilton-Jacobi方程的精神Fathi的方法马瑟理论。

拉格朗日动力学中的无穷粒子极限与最优输运无网格方法的收敛性

On the infinite particle limit in Lagrangian dynamics and convergence of optimal transportation meshfree methods

研究拉格朗日系统中无穷粒子极限下离散作用泛函的Gamma收敛性，并证明最优输运无网格方法对粒子流近似的收敛性。

A McKean optimal transportation perspective on Feynman-Kac formulae with application to data assimilation

Data assimilation is the task of combining mathematical models with observational data. From a mathematical perspective data assimilation leads to Bayesian inference problems which can be formulated in terms of Feynman-Kac formulae. In this paper we focus on the sequential nature of many data assimilation problems and their numerical implementation in form of Monte Carlo methods. We demonstrate how sequential data assimilation can be interpreted as time-dependent Markov processes, which is often referred to as the McKean approach to Feynman-Kac formulae. It is shown that the McKean approach ha…

Optimal Transportation for Generalized Lagrangian

In this paper, we study the optimal transportation for generalized Lagrangian $L=L(x, u,t)$, and consider the cost function as following: $$c(x, y)=\inf_{\substack{x(0)=x\\x(1)=y\\u\in\mathcal{U}}}\int_0^1L(x(s), u(x(s),s), s)ds.$$ Where $\mathcal{U}$ is a control set, and $x$ satisfies the following ordinary equation: $$\dot{x}(s)=f(x(s),u(x(s),s)).$$ We prove that under the condition that the initial measure $μ_0$ is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the correspondin…

给定压力场下的最优输运

Optimal Transportation in the presence of a prescribed pressure field

从几个角度讨论了最优（Monge-Kantorovich）运输问题。拉格朗日公式将{ \em拉格朗日} $L(v,x,t)$的作用量从$\R^n$中的轨道集扩展到一组测量值轨道。{ \em欧拉}对偶公式导致了相应的Hamilton-Jacobi方程的子解集上的优化问题。最后，通过将最优化问题归结为保测度映射和两点分布测度的集合，得到了Monge问题及其Kantorovich松弛.在本文中，我们专注于机械拉格朗日$L=|v|^2/2+P(x,t)$导致，在一般情况下，一个非齐次成本函数。主要结果产生存在一个独特的{ \em流}的同态运输的最佳措施值轨道的扩展拉格朗日，以及存在一个最佳的解决方案的对偶欧拉问题和它的关系的Monge和Kantorovich配方。

The intrinsic dynamics of optimal transport

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determini…