Mass transportation problems

doi:10.1007/B98893

Book•DOI•

Mass transportation problems

01 Jan 1998-

TL;DR: In this article, a modification of the Monge-Kantorovich Problem with relaxed or additional constraints is presented. But this modification is restricted to the case where the Kantorovich-type metrics are applied to various Probabilistic-Type Limit Theorems.

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Abstract: Modifications of the Monge-Kantorovich Problems: Transportation Problems with Relaxed or Additional Constraints.- Application of Kantorovich-Type Metrics to Various Probabilistic-Type Limit Theorems.- Mass Transportation Problems and Recursive Stochastic Equations.- Stochastic Differential Equations and Empirical Measures.

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Gradient Flows: In Metric Spaces and in the Space of Probability Measures

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Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré

01 Jan 2005

TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.

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Abstract: Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

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3,401 citations

Cites background from "Mass transportation problems"

...In the presentation of these facts we have been following mostly [14], [70], [111], [125]....
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Journal Article•DOI•

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

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Jean-David Benamou¹, Yann Brenier²•Institutions (2)

French Institute for Research in Computer Science and Automation¹, Institut Universitaire de France²

01 Jan 2000-Numerische Mathematik

TL;DR: The Monge-Kantorovich mass transfer problem is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

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Abstract: Summary. The $L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

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1,573 citations

Cites background or methods from "Mass transportation problems"

...A recent comprehensive review can be found in the new books by Rachev and R üschendorf [31], the lecture notes by Evans [19] and the review paper by Mc Cann and Gangbo [21]....
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...Examples of MKP and generalizations The assignment problem The dataρ0 and ρT can be much more general than bounded functions and probability measures can also be considered [31]....
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...Let us just briefly mention the theoretical importance of the L2 MKP in many different fields such as probability theory and statistics [31], functional analysis [3], kinetic theory (where theL2 Kantorovich distance is closely related to the homogeneous Boltzmann equation of maxwellian molecules and the Fokker-Planck equation [36,22]), atmospheric sciences (where the construction of the semigeostrophic model by Cullen and Purser is based on a variant of the L2 MKP [16,6]), astrophysics [26], porous media equations, Hele-Shaw equations (with the new approach introduced by Otto for dissipative PDEs viewed as gradient flows with respect to the L2 Kantorovich metric [28,29])....
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...Let us recall a basic theoretical result on the L2 MKP ([24,8,10], see also [31,21,19]): there is a unique optimal transfer M characterized as the unique map transferring ρ0 to ρT which can be written as the gradient of some convex functionΨ , M(x) = ∇Ψ(x)....
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...TheL2 Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method....
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Journal Article•DOI•

On the geometry of metric measure spaces. II

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Karl-Theodor Sturm¹•Institutions (1)

University of Bonn¹

01 Jul 2006-Acta Mathematica

TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.

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Abstract: We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

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1,382 citations

Monograph•DOI•

Geometry of Quantum States: Frontmatter

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Ingemar Bengtsson, Karol Życzkowski

01 Jan 2006

TL;DR: In this article, the space of isospectral 0Hermitian matrices is shown to be the space in which the number 6) and 7) occur twice in the figure, and the discussion between eqs.(5.14) and (5.15) is incorrect.

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Abstract: a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

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1,089 citations

Journal Article•DOI•

Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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Felix Otto¹, Cédric Villani²•Institutions (2)

University of Bonn¹, École Normale Supérieure²

01 Jun 2000-Journal of Functional Analysis

TL;DR: In this paper, it was shown that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6, 587-600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities.

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1,080 citations

Additional excerpts

...First of all, thanks to the triangle inequality for the Wasserstein metric (see [27] for instance), |W (μ, μt+s)−W (μ, μt)| ≤ W (μt, μt+s)....
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Mass transportation problems

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