论文标题
部分可观测时空混沌系统的无模型预测
Relaxed many-body optimal transport and related asymptotics
论文作者
论文摘要
$ \ Mathbb {r}^d $中概率度量的优化问题被认为是成本功能涉及多核心最佳运输的情况。在$ n $相互作用粒子的模型中，例如在密度函数理论中，相互作用成本是令人反感的，并且由两点函数$ c（x，y）= \ ell（| x-y |）$描述，其中$ \ ell：\ m athbb {r} _+ \ to [0，\ infty] $降低到Infinity at Infinination Infinity。由于可能存在无穷大的质量损失，因此可能会出现不存在的，而放松对亚概率的初始问题是必要的。在本文中，我们表征了\ cite {bouchitte2020raxed}的结果的放松功能，并提出了一种二重性方法，该方法允许根据成本$ \ ell（r）$在非常一般的假设下计算$γ-$限制为$ n \ to \ infty $。我们表明，该极限与所谓的直接能量的凸壳相吻合。然后，当应用连续的外部电位时，我们研究极限优化问题。给出条件，其中有明确的示例，其中最小化是概率或质量$ <1 $的情况。 In a last part we study the case of a small range interaction $\ell_N(r)=\ell (r/\varepsilon)$ ($\varepsilon\ll 1$) and we show how the duality approach can be also used to determine the limit energy as $\varepsilon\to 0$ of a very large number $N_\varepsilon$ of particles.
Optimization problems on probability measures in $\mathbb{R}^d$ are considered where the cost functional involves multi-marginal optimal transport. In a model of $N$ interacting particles, like in Density Functional Theory, the interaction cost is repulsive and described by a two-point function $c(x,y) =\ell(|x-y|)$ where $\ell: \mathbb{R}_+ \to [0,\infty]$ is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper we characterize the relaxed functional generalizing the results of \cite{bouchitte2020relaxed} and present a duality method which allows to compute the $Γ-$limit as $N\to\infty$ under very general assumptions on the cost $\ell(r)$. We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass $<1$ . In a last part we study the case of a small range interaction $\ell_N(r)=\ell (r/\varepsilon)$ ($\varepsilon\ll 1$) and we show how the duality approach can be also used to determine the limit energy as $\varepsilon\to 0$ of a very large number $N_\varepsilon$ of particles.
