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Canonical duality approach in the approximation of optimal Monge mass transfer mapping
TL;DR: In this article, a variational method was used to transform the 1-dimensional mass transfer problem into a sequence of minimization problems, which can then be converted into a nonlinear differential equation with constraints by applying the canonical duality method.
Abstract: This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping for the Monge problem according to the theoretical results.
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01 May 2003
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TL;DR: In this paper, the approximation of a global maximizer of the 1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary was investigated.
Abstract: This paper mainly investigates the approximation of a global maximizer of the 1-D Monge–Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge–Kantorovich problem will be demonstrated.
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TL;DR: In this paper, the existence and uniqueness of optimal maps are discussed. But the uniqueness of the optimal map is not discussed. And the role of the map in finding the optimal solution is left open.
Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal measures . . . . . . . . . . . . . . . . . . . 126 Part I. Strictly convex costs . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. Existence and uniqueness of optimal maps . . . . . . . . . . . . . 133 4. Characterization of the optimal map . . . . . . . . . . . . . . . . . 137 Part II. Costs which are strictly concave as a function of d i s t a n c e . . . 141 5. The role of optimal maps . . . . . . . . . . . . . . . . . . . . . . . . 141 6. Uniqueness of optimal solutions . . . . . . . . . . . . . . . . . . . . 144 Part III. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A. Legendre transforms and conjugate costs . . . . . . . . . . . . . . 148 B. Examples of c-concave potentials . . . . . . . . . . . . . . . . . . . 152 C. Regularity of c-concave potentials . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
939 citations
"Canonical duality approach in the a..." refers background in this paper
...Wang [5, 6, 7, 8, 19, 20, 29, 32], we provide another viewpoint, namely, nonlinear differential equation approach, for the Monge transfer problem....
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587 citations
"Canonical duality approach in the a..." refers background in this paper
...Clearly, like p−Laplacian, e 2 ε,y−α 2)/(2ε) is a highly nonlinear function, which is difficult to solve by the direct approach [3, 10, 24]....
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TL;DR: In this paper, uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the Transport density at the ends of rays Approximate mass transfer plans Passage to limits a.k.a. Optimality
Abstract: Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.
492 citations
"Canonical duality approach in the a..." refers background or methods in this paper
...Wang [5, 6, 7, 8, 19, 20, 29, 32], we provide another viewpoint, namely, nonlinear differential equation approach, for the Monge transfer problem....
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...Moser [6, 7, 8] provided an ODE recipe to build s by solving a flow problem involving...
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TL;DR: In this article, the authors introduced the notion of a generalized solution and proved the existence and uniqueness of generalized solutions of the optimal transportation problem, and also proved the solution is smooth under certain structural conditions on the cost function.
Abstract: The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampere type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.
411 citations
"Canonical duality approach in the a..." refers background in this paper
...Wang [5, 6, 7, 8, 19, 20, 29, 32], we provide another viewpoint, namely, nonlinear differential equation approach, for the Monge transfer problem....
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...There are many types of cost functions while dealing with different problems [1, 4, 7, 29]....
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...cǫ(x, y) = √ ǫ2 + |x− y|2 in the discussion of regularity [29, 32]....
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