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Multivariate Monge-Kantorovich Transportation Problem Analysis

Posted on:2010-07-30Degree:DoctorType:Dissertation
Country:ChinaCandidate:Y F ShenFull Text:PDF
GTID:1100360275494741Subject:Probability theory and mathematical statistics
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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 Rn; 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 R1, the above optimal transportation problem was completely solved, and the minimum of expected cost is∫01c(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.
Keywords/Search Tags:optimal transportation problem, Monge problem, Kantorovich problem, Monge-Kantorovich transportation problem, optimal mapping, optimal transfer plan, multivariate, quasi-linear, partial differential equaion
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