On the local structure of optimal measures in the multi-marginal optimal transportation problem
Brendan Pass
考虑了一个具有两个以上边值的最优运输问题。我们使用一个家庭的半黎曼度量来自混合，二阶偏导数的费用函数提供上界的支持的解决方案的尺寸。
We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.
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Uniqueness and Monge solutions in the multi-marginal optimal transportation problem
研究了一类多边际最优运输问题。在一定条件下的费用函数和第一个边际，我们证明了放松，Kantorovich版本的问题的解决方案诱导蒙赫问题的解决方案，这两个问题的解决方案是唯一的。
Optimal transportation with infinitely many marginals
我们提出并研究了一个具有无穷多个边际的最优运输问题;这是Gangbo和Swiech（1998）研究的多边际问题的自然扩张。我们证明了优化器的存在性、唯一性和特征，这些结果是Gangbo和Swiech结果的自然扩张。这个证明依赖于这个问题和在Wasserstein空间中寻找重心的问题之间的关系，这是Agueh和Carlier（2011）首次在许多边缘上观察到的联系。
Regularity of optimal transportation between spaces with different dimensions
研究了一类最优运输问题解的正则性，其中源的维数大于目标的维数。我们证明了，如果目标是$c$ -凸的，那么源有一个规范的叶状，其余维等于目标的尺寸和问题减少到一个最佳的运输问题之间的空间具有相同的尺寸。如果$c$ -凸性条件失败，我们不期望任何光滑边缘的正则性，但是，在源是2维和目标是1维的情况下，我们确定边缘和成本的充分条件，以确保最佳映射是连续的。
On a class of optimal transportation problems with infinitely many marginals
我们证明了一类最优运输问题的解的存在性和唯一性结果，无穷多个边缘，支持在真正的线。我们还提供了一个明确的公式的解决方案的特征。然后，我们证明了这个结果意味着一个无穷维的重排不等式，并使用它来获得上界的解决方案，抛物型偏微分方程通过Feynman-Kac公式。这一结果可以用来提供数学金融中某些衍生品价格的模型独立边界。作为另一个应用，我们得到了量子物理中的精细相空间边界。
New Computable Necessary Conditions for the Regularity Theory of Optimal Transportation
给出了一类最优运输问题有光滑解的新的可计算必要条件。
Generic uniqueness of optimal transportation networks
本文基于Baire范畴证明了对于通用边界，最优分支运输问题存在唯一的极小化子，解决了最优运输网络的通用唯一性问题。
On the degeneracy of optimal transportation
我们扩展了一个维度上界多少最优输运地图可以退化的二次运输成本，最初是由于Caffarelli，费用函数，满足曲率条件的马，Trudinger和王。
The Monge-Ampère equation and its link to optimal transportation
我们调查（旧的和新的）正则性理论的蒙日-安培方程，显示其连接到最佳运输，并描述了正则性的一般类蒙日-安培型方程在这种情况下产生的。
Rectifiability of Optimal Transportation Plans
本文的目的是证明Kantorovich最优运输问题的解是在Lipschitz流形上支持的，只要费用是具有非奇异混合二阶导数的$C^{2}$.我们使用这个结果提供一个简单的证明，解决蒙日的最佳运输问题满足变量变换方程几乎处处。
We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on $C^2$ uniformly convex domains are $C^{1,α}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.
We collect some examples of optimal transports in order to explore the (in)stability of the identity map as an optimal transport. First, we consider density and domain perturbations near regular portions of domains. Second, we investigate density and domains deformations to non-regular parts of domains. Here, we restrict our attention to two dimensions and focus near 90 degree corners.
In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures $μ^-$, $μ^+$ on $\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$ with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph $\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in $\mathbb{R}^m$. The constraint is encoded in the relation ${\rm div}(T)=μ^--μ^+$. The cost…
We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Amp{è}re equation, which is based on the maximum principle, our approach is purely variational. By constructing a competitor on the level of the Eulerian (Benamou-Brenier) formulation, we show that locally, the velocity is close to the gradient of a harmonic function provided the transportation cost is small. We then translate back to the Lagrangian description and perform a Campanato iteration to obtain an $ε$…
We show in full generality the stability of optimal traffic paths in branched transport: namely we prove that any limit of optimal traffic paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks, by Bernot, Caselles and Morel), which has been addressed up to now only under restrictive assumptions.
Multi-marginal optimal transport on Riemannian manifolds
研究了黎曼流形上的多边际最优运输问题，其费用函数由重点点到重心的平均距离平方给出.在第一个边际上的标准正则性条件下，我们证明了最优测度是唯一的，并且集中在第一个变量上的函数的图上，从而导出了Monge解。这个结果推广了McCann关于流形的极因子分解定理，从2到几个边值，在同样的意义上，一个著名的结果Gangbo和Swiech推广了Brenier的极因子分解定理在$\mathbb{R}^n$。
A parabolic flow toward solutions of the optimal transportation problem on domains with boundary
我们考虑一个抛物线版本的质量运输问题，并表明，它收敛到一个解决方案的原始质量运输问题在适当的条件下的费用函数，初始和目标域。
In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,α}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^α$, and $\partialΩ, \partialΩ^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampère equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,α}$. As an application, in the last we also obtain these regularities of the free…
Regularity properties of optimal transportation problems arising in hedonic pricing models
本文研究了享乐定价模型中的最优运输剩余函数。我们导出了这些函数的Ma-Trudinger-Wang曲率公式，给出了它们满足\textbf{(A3w)}的充要条件.我们用这个给明确的新的例子，盈余函数满足\textbf{(A3w)}，形式$b(x,y)=H(x+y)$其中$H$是一个凸函数$\mathbb{R}^n$。我们还表明，在享乐定价模型的均衡合同的空间具有最大可能的尺寸，潜在的经济利益的结果。
On strict convexity and $C^1$ regularity of potential functions in optimal transportation
本文讨论了最优运输中费用函数和区域上的条件与位势的凸性之间的关系以及相应的最优映射的连续性。特别地，我们证明了，如果费用函数满足条件（A3），在我们以前的工作与Xinan马，密度和他们的倒数是有界的和目标域是凸的费用函数，那么潜力是连续可微的和它的对偶势严格凹的费用函数。我们的结果延伸，不同的和更直接的证明，类似的结果Loeper证明近似从我们早期的工作的整体正则性。
This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Amp{è}re equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.
$C^{2,α}$ regularity of free boundaries in optimal transportation
本文通过建立自由边界的几何估计并研究Monge–Ampère方程第二边值问题，证明最优运输中势函数与自由边界均具有$C^{2,\alpha}$正则性，解决了Caffarelli–McCann公开问题。
An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation
本文在整个空间$H^1$上构造了Camassa-Holm方程的整体连续解流。我们的解是保守的，在这个意义上，总能量$\int (u^2+u_x^2) dx$保持a.e.在时间上恒定。我们的新方法是基于距离泛函$J(u,v)$，定义在一个最优运输问题，满足${d\over dt} J(u(t), v(t))\leq κ\cdot J(u(t),v(t))$的每对夫妇的解决方案。利用这个新的距离泛函，我们可以构造任意解作为多峰子解的一致极限，并证明了一个普遍的唯一性结果。
Hölder and Sobolev regularity of optimal transportation potentials with rough measures
本文研究粗糙测度间最优传输的 Kantorovich 势函数，在球上与 Lebesgue 测度可比的条件下，证明其至依赖 δ 的临界尺度具有 Hölder 与 Sobolev 正则性。
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result u…
Regularity of singular set in optimal transportation
本文研究目标为两个不相交凸区域的最优运输问题，在不要求源区域凸性的条件下，通过新方法证明了奇异集的高阶正则性，与Caffarelli的经典理论相呼应。
On the second boundary value problem for Monge-Ampere type equations and optimal transportation
本文研究Monge-Ampere方程第二边值问题整体光滑解的存在性及其在最优运输中势函数正则性的应用.费用函数满足条件A3的一个弱形式，在此条件下，我们在最近的一篇文章中证明了内部正则性。因此，它们包括Caffarelli和Urbas的二次费用函数情况以及早期工作中的各种示例。该方法是通过推导解决方案的二阶导数的全球估计。
Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions
我们证明了唯一性和Monge解决方案的结果多边际最优运输问题与一定类别的盈余功能，这类自然出现在经济学中的多智能体匹配问题。这一结果推广了Gangbo和Schwarzwich关于多边际问题的一个开创性结果。特别感兴趣的是，我们表明，这也产生了部分推广的Gangbo-Eschwiffich结果流形;或者，我们可以认为这是一个部分扩展的麦肯定理的二次成本流形的多边际设置。我们还证明了本文所考虑的盈余函数类既不包含也不包含在作者所研究的另一类盈余函数类中，这也推广了Gangbo和Schuwienrich的结果.
Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds
给出了截面曲率处处为正的黎曼流形上最优输运连续性的反例。这些例子表明Ma，Trudinger，& Wang的条件A3w不能由截面曲率的正性保证。
Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures
研究了将一个概率测度$μ= e^{-V} \ dx$推进到另一个概率测度$ν= e^{-W} \ dx$上的最优运输映射$\nabla Φ: \mathbb{R}^d \mapsto \mathbb{R}^d$.按照E. Calabi在$\mathbb{R}^d$上引入了黎曼度量$g = D^2 Φ$，并研究了度量测度空间$M=(\mathbb{R}^d, g, μ)$的谱性质.我们证明，特别是，$M$承认一个非负的Bakry-{\displaystyle {\frac} merry张量提供$V$和$W$是凸的。若目标测度$ν$是凸集$Ω$上的Lebesgue测度，且$μ$是对数凹的，则证明了$M$是$CD(K,N)$空间.这些结果的应用包括$\| D^2 Φ\|$的一些全局无量纲先验估计。借助于黎曼流形上的比较技巧和概率集中论，我们证明了$M$的一些直径估计.
On the regularity of maps solutions of optimal transportation problems
给出了一个关于代价函数的充要条件，使得Monge最优运输问题的映射解对任意光滑正数据是连续的.该条件首先由Ma，Trudinger和Wang \cite{MTW, TW}引入，用于相应的蒙日-安培方程的先验估计。它由非负的所谓{ \em成本截面曲率}表示。我们证明了当代价函数是黎曼流形的平方距离时，代价截面曲率产生截面曲率。因此，如果流形不是处处具有非负截面曲率，则对于任意光滑正数据，最优输运映射{ \em不可能是连续的}。成本截面曲率的非负性等价于任何成本凸函数（凸函数的适当推广）与其任何支撑函数之间的接触集的连通性。当成本截面曲率是一致正的，我们得到的最佳映射是连续的或Hölder连续的数据相当弱的假设下，相比，需要在欧几里得的情况下。这种情况包括反射器天线问题和球面上的平方黎曼距离。
In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. This covers for instance, $N$-bo…
Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere
给出了初始测度和目标测度在球面$§^n$上的充分条件，以保证费用为$|x-y|^2/2$的最优输运问题的解是同构的.
Pseudo-Riemannian geometry calibrates optimal transportation
给定运输成本$c: M \times\bar M \to\mathbf{R}$，最优映射最小化将质量从$M$移动到$\bar M$的总成本。我们在$M\times\bar M$上找到了一个伪度量和一个校准形式，使得最优映射的图是一个校准极大子流形。我们定义了空间中的类空电流的质量与不定度规。
On global Hölder estimates for optimal transportation
推广了L.一致对数凹概率测度间最优运输$T$的Caffarelli-Lipschitz估计设$T : \R^d \to \R^d$是将$μ= e^{-V}dx$向前推到$ν= e^{-W}dx$的最优传输。假设1）$V$的二阶微商可以通过幂函数从上估计，2）$W$的凸性模可以通过$A_q |x|^{1+q}$、$q \ge 1$从下估计。在这些假设下，我们证明了$T$是具有无量纲系数的全局Hölder。此外，我们还研究了有界凸集$K \subset \R^d$上$μ$与一致测度之间的最优迁移$T$。我们用$d$，${diam(K)}$和$D V, D^2 V$得到了$T$的Lipschitz常数的估计.
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。
Limit Theorems for Optimal Mass Transportation
最优质量运输是由蒙日介绍了约200年前，是，今天，来源的大量结果分析，几何和凸性。在这里，我调查了一个新的，令人惊讶的最佳变换之间的联系，通过不同的拉格朗日行动黎曼流形。作为一种特殊情况，对于任何一对质量相等的非负测度$λ^+,λ^-$ $$ W_1(λ^-, λ^+)= \lim_{\eps\to 0} \eps^{-1}\inf_μ W_p(μ+\epsλ^-, μ+\epsλ^+)$$，其中$W_p$，$p\geq 1$是Wasserstein距离，下确界在周围空间中的概率测度集合上。
This paper studies a variant of ramified/branched optimal transportation problems. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $μ$ and $ν$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{α}(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tildeμ$ to $\tildeν$ with $\tildeμ\leq μ$ and $\tildeν\leq ν$, where $\mathbf{M}_…
On the Regularity of Optimal Transportation Potentials on Round Spheres
本文研究了定义在圆球上的最优输运势的正则性。具体来说，这项研究概括了Loeper所做的计算，他证明了Trudinger和Wang的强（A3）条件在圆形球体上得到满足，当成本函数是测地距离平方时。为了将Loeper的计算推广到更广泛的一类成本函数，（A3）条件通过球极平面投影重新表述，该投影将球体的图表映射到欧几里得空间。这种重新表述随后允许验证任何情况下的（A3）条件，其中相关的最优运输问题的成本函数可以表示为一个函数的测地线之间的距离在一个圆形的球体。有了这个，几个例子，这样的成本函数，然后分析，看看他们是否满足这个（A3）条件。
Five lectures on optimal transportation: Geometry, regularity and applications
在这一系列讲座中，我们介绍了蒙格-康托洛维奇问题的最佳运输一个分布的质量到另一个，其中最优性是衡量对费用函数c（x，y）。连接到几何，不等式，和偏微分方程将被讨论，特别是集中在最近的发展，在正则性理论的蒙赫安培型方程。本文还将介绍一个在微观经济学中的应用，即寻找一个垄断者的均衡价格分布，该垄断者向一群匿名代理人销售一系列多维产品，这些代理人的偏好仅在统计学上是已知的。
In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well. In particular, in the case f = 1 (projection onto a set of densities with an L^\infty bound) we precisely prove that the total variation of the projection does not exceed the total …
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,α}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoo…
A regularized transportation cost stemming from entropic approximation
本文研究了在熵正则化下最优运输问题的收敛性，通过弱紧致性条件证明正则化函数的收敛，并探讨了多边际问题的应用。
An iterative scheme for solving the optimal transportation problem
针对离散目标测度的最优运输问题，提出一种迭代近似方案，并给出迭代终止所需的有限上界。
Some reverse inequality in optimal mass transportation
本文研究了最优质量运输中点wise成本递减函数下的反向不等式，提出了一种通用框架以统一不同成本函数的情况。
Eigenvalue distribution of optimal transportation
我们研究了$\mathbb{R}^n$中两个凸区域上的一致测度之间的Brenier映射$\nabla Φ$，或者更一般地，研究了$\mathbb{R}^n$上两个对数凹概率测度之间的Brenier映射$\nabla Φ$.我们表明，黑塞矩阵$D^2 Φ$的特征值在乘法尺度上表现出显着的浓度特性，无论选择的两个措施或尺寸$n$。
This paper is devoted to theoretical and numerical investigation of the local minimum issue in seismic full waveform inversion (FWI). This paper provides a mathematical analysis of optimal transportation (OT) type objective function's differentiability and proves that the gradient obtained in the adjoint-state method does not depend on the particular choice of the Kantorovich potentials. A novel approach using the softplus encoding method is presented to generalize and impose the OT metric on FWI. This approach improves the convexity of the objective function and mitigates the cycle-skipping p…
We introduce and study a new optimal transport problem on a bounded domain $\barΩ\subset \mathbb R^d$, defined via a dynamical Benamou-Brenier formulation. The model handles differently the motion in the interior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on $\barΩ$ on the one hand, and on the other hand between two independent optimal transport problems set on $Ω$ and $\partial Ω$.
We show that in any complete metric space the probability measures $μ$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $ν$ living on the support of $μ$ is bounded below by a positive function of the $L^\infty$ transportation distance between $μ$ and $ν$. The function giving the lower bound depends only on the lower bound of the $μ$-measures of balls centered at the support of $μ$ and on the cost function used in the optimal transport. We obtain an essentially sharp form of this function. In the case of…
Density functional theory and optimal transportation with Coulomb cost
本文基于最优输运的观点，对密度泛函理论中的交换关联泛函进行了新的研究。我们表明，在两个电子的情况下，在半经典极限，确切的交换相关功能减少到一个非常有趣的新形式的功能，这取决于一个最优输运映射$T$与一个给定的密度$ρ$。由于上述极限是强相关的，极限泛函产生洞察电子相关性。我们证明了这样的最佳映射的存在性和唯一性的任何数量的电子和每个$ρ$，并确定明确的情况下，当$ρ$是径向对称的地图。
A viscosity solution approach to the Monge-Ampere formulation of the Optimal Transportation Problem
本文提出一种通过求解完全非线性椭圆偏微分方程（Monge-Ampere方程）的第二边值问题来数值求解最优质量传输问题的方法，并利用粘性解理论证明了收敛性。
Mass transportation and contractions
根据L. Caffarelli证明了将标准高斯测度推进到对数凹测度$e^{-W} dx$上且满足$D^2 W \ge {Id}$的最优质量输运映射是1-Lipschitz。我们提出了一个简短的调查相关的结果和各种应用。
In this paper we analyze a mass transportation problem in a bounded domain with the possibility to transport mass to/from the boundary, paying the transport cost, that is given by the Euclidean distance plus an extra cost depending on the exit/entrance point. This problem appears in import/export model, as well as in some shape optimization problems. We study the L^p summability of the transport density which does not follow from standard theorems, as the target measures are not absolutely continuous but they have some parts of them which are concentrated on the boundary. We also provide the r…
We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Carathéodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.
We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem f…
Optimal Transportation Theory with Repulsive Costs
本文探讨了具有多个边际的最优运输理论在排斥成本函数中的最新发展，结合密度泛函理论从数学角度重新审视最优运输理论，并介绍了最近在库仑成本、排斥谐波成本和行列式成本方面的理论和数值结果。
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。
In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $\R^2$. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.
Numerical solution of the Optimal Transportation problem using the Monge-Ampere equation
提出一种结合最优传输边界条件的Monge-Ampère方程有限差分法，通过牛顿迭代实现快速求解，并支持非凸和非连通域上的密度计算。
Iterative scheme for solving optimal transportation problems arising in reflector design
针对反射器设计中的最优传输问题，提出一种迭代线性规划方案，通过逐步减少约束数量实现更精细网格的数值求解。
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