Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem

Pass, Brendan

doi:10.1137/100804917

Cited by 69 publications

(83 citation statements)

References 22 publications

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“…For continuous problems with N ≥ 3, whether the ansatz (1.7) works appears to depend on subtle properties of the cost function and even on the ambient space dimension; for the Coulomb cost with X = R 3 it is presently unknown. See [GS98,He02,Ca03,Pa11,CDD13] for interesting examples where minimizers of (1.1) are of Monge form, with the first result of this type appearing in a fundamental paper by Gangbo and Święch [GS98]. Examples of non-Monge minimizers can be found in [Pa10,FMPCK13,Pa13], and see [MP17] for an example with unique non-Monge minimizer.…”

Section: Introductionmentioning

confidence: 99%

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

Friesecke¹,

Vögler²

2018

SIAM J. Math. Anal.

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We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from N + −1 −1 to · (N + 1), where is the number of marginal states and N the number of marginals.The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to · (N − 1) unknowns, and cures the insufficiency of the Monge ansatz, i.e. we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions.Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context N corresponds to the number of particles, motivating the interest in large N .

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Section: Introductionmentioning

confidence: 99%

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

Friesecke¹,

Vögler²

2018

SIAM J. Math. Anal.

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“…This is then applied to obtain some estimates of the cost and to the study of continuity properties.This problem is called a multimarginal optimal transportation problem and elements of Π(ρ) are called transportation plans for ρ. Some general results about multimarginal optimal transportation problems are available in [3,17,21,22,23]. Results for particular cost functions are available, for example in [11] for the quadratic cost, with some generalization in [15], and in [4] for the determinant cost function.Optimization problems for the cost function C(ρ) in (1.1) intervene in the socalled Density Functional Theory (DFT), we refer to [16,18] for the basic theory of DFT and to [13,14,24,25,26] for recent development which are of interest for us.…”

mentioning

confidence: 99%

“…This problem is called a multimarginal optimal transportation problem and elements of Π(ρ) are called transportation plans for ρ. Some general results about multimarginal optimal transportation problems are available in [3,17,21,22,23]. Results for particular cost functions are available, for example in [11] for the quadratic cost, with some generalization in [15], and in [4] for the determinant cost function.…”

mentioning

confidence: 99%

Continuity and Estimates for Multimarginal Optimal Transportation Problems with Singular Costs

2017

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We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.This problem is called a multimarginal optimal transportation problem and elements of Π(ρ) are called transportation plans for ρ. Some general results about multimarginal optimal transportation problems are available in [3,17,21,22,23]. Results for particular cost functions are available, for example in [11] for the quadratic cost, with some generalization in [15], and in [4] for the determinant cost function.Optimization problems for the cost function C(ρ) in (1.1) intervene in the socalled Density Functional Theory (DFT), we refer to [16,18] for the basic theory of DFT and to [13,14,24,25,26] for recent development which are of interest for us. Some new applications are emerging for example in [12]. In the particular case of the Coulomb cost there are also many other open questions related to the applications. Recent results on the topic are contained in [2,5,6,7,10,20] and some of them will

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“…We will use a heuristic refinement mesh strategy allowing to obtain more accuracy without increasing the computational cost and memory requirements. This idea was introduced in [27] for the adaptative resolution of the pure Linear Programming formulation of the Optimal Transportation problem, i.e without the entropic regularisation.…”

Section: A Heuristic Refinement Mesh Strategymentioning

confidence: 99%

A Numerical Method to Solve Multi-Marginal Optimal Transport Problems with Coulomb Cost

Benamou

Carlier

Nenna

2016

Scientific Computation

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In this paper, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is to introduce an entropic regularization of the Kantorovich formulation of the Optimal Transport problem. The regularized problem then corresponds to the projection of a vector on the intersection of the constraints with respect to the Kullback-Leibler distance. Iterative Bregman projections on each marginal constraint are explicit which enables us to approximate the optimal transport plan. We validate the numerical method against analytical test cases.

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Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem

Cited by 69 publications

References 22 publications

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

Continuity and Estimates for Multimarginal Optimal Transportation Problems with Singular Costs

A Numerical Method to Solve Multi-Marginal Optimal Transport Problems with Coulomb Cost

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