多元Monge-Kantorovich运输问题研究
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摘要
1781年,Monge,G.提出了一个最优运输问题(即Monge问题),考虑把一定量的沙子从一地运到另一地,找到使总的运输费用最小的最优途径(数学上又称为最优映射)。1942年Kantorovich,L.V.也对此类运输问题提出了数学构造。他取出发点和终点的容积分别是度量空间上给定的两个概率测度,考虑如何找到以这两个概率测度为边际的联合概率测度(又称为最优运输计划),使得两者之间的运输费用最小。具体构造为:给定R~n上的概率测度P,Q,一个以P,Q为边际分布的2n维随机向量(X,Y)称为P,Q的耦合;给定一个费用函数c(·,·),我们要对所有的耦合最小化期望费用E[c(X,Y)]。因此人们又把上述运输问题统称为Mong-Kantorovich运输问题。
在R~1,上述运输问题早已完全解决,期望费用最小值为∫_0~1c(F~(-1)(t)－G~(-1)(t))dt,其中F,G分别是P,Q的分布函数,F~(-1),G~(-1)分别是F,G的右逆。但多元情形很久没有突破性进展。直到1991年Brenier,Y.用凸函数的梯度刻画了最优映射,从而将单位费用c(x,y)=|x-y|~2(|x-y|表示x,y之间的欧几里得距离,下同)时Monge-Kantorovich运输问题与经典的偏微分方程—Monge-Amp(?)re方程联系起来了。他的这篇文章建立了运输问题与偏微分方程、流体力学、几何学、概率论与泛函分析的美妙联系。从此,Monge-Kantorovich运输问题变得极其流行,因为许多不同领域的学者意识到这个主题与他们的领域联系非常密切,尤其引起了一大批偏微分方程方面专家学者的浓厚兴趣,如Evans,Trudinger分别得到在对已知测度的一定假设条件下,最优映射所满足的偏微分方程,其他大部分学者如Caffarelli等则用Monge-Kantorovich运输问题来研究一些经典的偏微分方程的解的有关性质。
然而,他们的证明方法都比较复杂,所得到的偏微分方程也不太容易处理和实际应用。本文考虑当单位费用函数c(x,y)=|x-y|~p(p≥2)时的多元Monge-Kantorovich运输问题(本文中简称为p-Monge-Kantorovich运输问题)。我们从概率论的角度,结合变分法,将多元p-Monge-Kantorovich运输问题转化成求解偏微分方程或偏微分方程组问题。特别地将二元2-Monge-Kantorovich运输问题(本文中也简称为平方Monge-Kantorovich运输问题)转化为一个Dirichlet边界的拟线性椭圆方程其中A(·,·)>0,B(·,·)>0,C是由初始分布决定的函数,H是未知的分布函数,而且我们得到了最优映射的显式表达式。同时我们讨论了离散情形,这样也就得到了Monge-Amp(?)re方程的一个数值计算方法。最后我们分析了p-Monge-Kantorovich运输问题(p>2)。显然此方法也适用于一般的凸单位费用函数。
本文将分成如下几部分进行阐述。
第一部分,回顾Monge-Kantorovich运输问题的现有主要结果,同时给出本文的主要研究思路。
第二部分,从概率论的角度,利用变分法于多元平方Monge-Kantorovich运输问题的连续情形。二元时我们得到一个Dirichlet边界的拟线性椭圆方程,更高维时得到一个偏微分方程组。同时我们得到了最优映射的显式表达式。
第三部分,考虑格子点空间上的最优耦合,给出多元离散平方Monge-Kantorovich运输问题最优解的刻画,从而也就得到了Monge-Amp(?)re方程的一个数值计算方法。
第四部分,进一步考虑p-Monge-Kantorovich运输问题(p>2),也得到一个偏微分方程。
The optimal transportation problem (called Monge Problem) was first formulated by Monge, G. in 1781, and concerned finding the optimal way, in the sense of minimal total transportation cost of moving a pile of soil from one site to another (in mathematics, also be called optimal mapping). This problem was given a modern mathematical formulation in the work of Kantorovich, L.V. in 1942 without knowledge of Monge, and so is now known as the Monge-Kantorovich transportation problem, where the initial mass and the final mass can be considered as probability measures on a metric space. The mathematical formulation is: Given two probability measures P and Q on R~n; a 2n dimensional random vector (X, Y) with P and Q as their respective marginal distributions is called a coupling of this pair (P, Q); Given a cost function c(·,·), we are interested in minimizing the expected cost E[c(X, Y)] among all possible couplings.
In R~1, the above optimal transportation problem was completely solved, and the minimum of expected cost is∫_0~1c(F~(-1)(t)－G~(-1)(t))dt, where F, G is the distributions of P,Q, F~(-1),G~(-1) is the right inverses of F, G, respectively. However, it took long time to make a real breakthrough in higher dimension. Until 1991 noted by Brenier, Y., who characterized the optimal maps in terms of gradients of convex functions, and thus connected the Monge-Kantorovich transportation problem as unit cost c(x, y) = |x—y|~2 with the classical partial differential equation-Monge-Ampere equation, where |x - y| denotes the Euclidean distance between x, y. His paper paved the way towards a beautiful interplay between partial differential equations, fluid mechanics, geometry, probability theory and functional analysis. Prom then on, it gained extreme popularity, because many researchers in different areas of mathematics understood that this topic was strongly linked to their subjects. Particulary, lots of specialists in partial differential equation field have started to be very interested in this problem. For example, under some assumptions on the given measures, Evans, Trudinger have obtained the special partial differential equations that the optimal mappings have been satisfied. And most authors such as Caffarelli and etc. studied properties of the solutions of some classical partial differential equations based on Monge-Kantorovich transportation problem.
However, their proof methods are very complex, also it is not very easy to handle the partial differential equations they've obtained and apply them into practices. In this text, we consider multivariate Monge-Kantorovich transportation problem as cost functions c(x,y) = |x-y|~p (here called p-Monge-Kantorovich transportation problem), p≥2. We use calculus of variations from probability point of view to transfer the bivariate transportation problem as p = 2 (here also called quadratic Monge-Kantorovich transportation problem) to Dirichlet problem associated to a quasi-linear elliptic equationwhere A(·,·) > 0,B(·,·) > 0, C is decided by the given distributions, H is an unknown distribution function, more, we have obtained the explicit formula of optimal map. Meanwhile, We also apply this method into the discrete cases, so we get another numerical calculation method of Monge-Amp(?)re equation. Finally, we consider p-Monge-Kantorovich transportation problem (p > 2). Of course, our method also holds for general convex cost functions.
The full text is totally divided into the following parts to go on.
In the first part, we review the main existing results about Monge-Kantorovich transportation problem. Also we put forward the main research way in our text.
In the second part, we apply our method into multivariate quadratic Monge-Kantorovich transportation problem in continuous cases. We get a Dirichlet problem associated to a quasi-linear elliptic equation in two dimension, and a partial differential equation system in higher dimension. And we have obtained the explicit formula of optimal map.
In the third part, we consider the optimal coupling on the lattice spaces, and show the characterizations of optimal solutions of multivariate quadratic Monge-Kantorovich transportation problem cases in discrete cases.
In the fourth part, we further consider p-Monge-Kantorovich transportation problem (p > 2), and also get a partial differential equation.
在R~1,上述运输问题早已完全解决,期望费用最小值为∫_0~1c(F~(-1)(t)－G~(-1)(t))dt,其中F,G分别是P,Q的分布函数,F~(-1),G~(-1)分别是F,G的右逆。但多元情形很久没有突破性进展。直到1991年Brenier,Y.用凸函数的梯度刻画了最优映射,从而将单位费用c(x,y)=|x-y|~2(|x-y|表示x,y之间的欧几里得距离,下同)时Monge-Kantorovich运输问题与经典的偏微分方程—Monge-Amp(?)re方程联系起来了。他的这篇文章建立了运输问题与偏微分方程、流体力学、几何学、概率论与泛函分析的美妙联系。从此,Monge-Kantorovich运输问题变得极其流行,因为许多不同领域的学者意识到这个主题与他们的领域联系非常密切,尤其引起了一大批偏微分方程方面专家学者的浓厚兴趣,如Evans,Trudinger分别得到在对已知测度的一定假设条件下,最优映射所满足的偏微分方程,其他大部分学者如Caffarelli等则用Monge-Kantorovich运输问题来研究一些经典的偏微分方程的解的有关性质。
然而,他们的证明方法都比较复杂,所得到的偏微分方程也不太容易处理和实际应用。本文考虑当单位费用函数c(x,y)=|x-y|~p(p≥2)时的多元Monge-Kantorovich运输问题(本文中简称为p-Monge-Kantorovich运输问题)。我们从概率论的角度,结合变分法,将多元p-Monge-Kantorovich运输问题转化成求解偏微分方程或偏微分方程组问题。特别地将二元2-Monge-Kantorovich运输问题(本文中也简称为平方Monge-Kantorovich运输问题)转化为一个Dirichlet边界的拟线性椭圆方程其中A(·,·)>0,B(·,·)>0,C是由初始分布决定的函数,H是未知的分布函数,而且我们得到了最优映射的显式表达式。同时我们讨论了离散情形,这样也就得到了Monge-Amp(?)re方程的一个数值计算方法。最后我们分析了p-Monge-Kantorovich运输问题(p>2)。显然此方法也适用于一般的凸单位费用函数。
本文将分成如下几部分进行阐述。
第一部分,回顾Monge-Kantorovich运输问题的现有主要结果,同时给出本文的主要研究思路。
第二部分,从概率论的角度,利用变分法于多元平方Monge-Kantorovich运输问题的连续情形。二元时我们得到一个Dirichlet边界的拟线性椭圆方程,更高维时得到一个偏微分方程组。同时我们得到了最优映射的显式表达式。
第三部分,考虑格子点空间上的最优耦合,给出多元离散平方Monge-Kantorovich运输问题最优解的刻画,从而也就得到了Monge-Amp(?)re方程的一个数值计算方法。
第四部分,进一步考虑p-Monge-Kantorovich运输问题(p>2),也得到一个偏微分方程。
The optimal transportation problem (called Monge Problem) was first formulated by Monge, G. in 1781, and concerned finding the optimal way, in the sense of minimal total transportation cost of moving a pile of soil from one site to another (in mathematics, also be called optimal mapping). This problem was given a modern mathematical formulation in the work of Kantorovich, L.V. in 1942 without knowledge of Monge, and so is now known as the Monge-Kantorovich transportation problem, where the initial mass and the final mass can be considered as probability measures on a metric space. The mathematical formulation is: Given two probability measures P and Q on R~n; a 2n dimensional random vector (X, Y) with P and Q as their respective marginal distributions is called a coupling of this pair (P, Q); Given a cost function c(·,·), we are interested in minimizing the expected cost E[c(X, Y)] among all possible couplings.
In R~1, the above optimal transportation problem was completely solved, and the minimum of expected cost is∫_0~1c(F~(-1)(t)－G~(-1)(t))dt, where F, G is the distributions of P,Q, F~(-1),G~(-1) is the right inverses of F, G, respectively. However, it took long time to make a real breakthrough in higher dimension. Until 1991 noted by Brenier, Y., who characterized the optimal maps in terms of gradients of convex functions, and thus connected the Monge-Kantorovich transportation problem as unit cost c(x, y) = |x—y|~2 with the classical partial differential equation-Monge-Ampere equation, where |x - y| denotes the Euclidean distance between x, y. His paper paved the way towards a beautiful interplay between partial differential equations, fluid mechanics, geometry, probability theory and functional analysis. Prom then on, it gained extreme popularity, because many researchers in different areas of mathematics understood that this topic was strongly linked to their subjects. Particulary, lots of specialists in partial differential equation field have started to be very interested in this problem. For example, under some assumptions on the given measures, Evans, Trudinger have obtained the special partial differential equations that the optimal mappings have been satisfied. And most authors such as Caffarelli and etc. studied properties of the solutions of some classical partial differential equations based on Monge-Kantorovich transportation problem.
However, their proof methods are very complex, also it is not very easy to handle the partial differential equations they've obtained and apply them into practices. In this text, we consider multivariate Monge-Kantorovich transportation problem as cost functions c(x,y) = |x-y|~p (here called p-Monge-Kantorovich transportation problem), p≥2. We use calculus of variations from probability point of view to transfer the bivariate transportation problem as p = 2 (here also called quadratic Monge-Kantorovich transportation problem) to Dirichlet problem associated to a quasi-linear elliptic equationwhere A(·,·) > 0,B(·,·) > 0, C is decided by the given distributions, H is an unknown distribution function, more, we have obtained the explicit formula of optimal map. Meanwhile, We also apply this method into the discrete cases, so we get another numerical calculation method of Monge-Amp(?)re equation. Finally, we consider p-Monge-Kantorovich transportation problem (p > 2). Of course, our method also holds for general convex cost functions.
The full text is totally divided into the following parts to go on.
In the first part, we review the main existing results about Monge-Kantorovich transportation problem. Also we put forward the main research way in our text.
In the second part, we apply our method into multivariate quadratic Monge-Kantorovich transportation problem in continuous cases. We get a Dirichlet problem associated to a quasi-linear elliptic equation in two dimension, and a partial differential equation system in higher dimension. And we have obtained the explicit formula of optimal map.
In the third part, we consider the optimal coupling on the lattice spaces, and show the characterizations of optimal solutions of multivariate quadratic Monge-Kantorovich transportation problem cases in discrete cases.
In the fourth part, we further consider p-Monge-Kantorovich transportation problem (p > 2), and also get a partial differential equation.
引文
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[2] Ambrosio, L., Pratelli, A.: Existence and stability results in the L~1 theory of optimal transportation. To appear in the proceedings of the CIME course held in Martina Franca
(Italy), "Optiaml transportation and application", 2-9 September 2001, Lecture Notes in Mathematics.
[3] Ambrosio, L.: Lecture notes on transport problems. In 'Mathematical Aspects of Evolving Interfaces', Lecture Notes in Mathematics, 1812, 1-52, Springer, Berlin, 2003.
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