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

By Jean-David Benamou, Yann Brenier. 2000

2,008 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

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Citations in the corpus

Equivariant Flows: exact likelihood generative learning for symmetric densities (2020, background): “Gradient flows (not necessarily G-equivariant) can map any ρ onto any ρ′ over Rn as long as both densities do not vanish (Benamou & Brenier, 2000; McCann, 2001) and have been discussed in the context of density estimation (Zhang et al., 2018; Papamakarios et al., 2019).”
Regularity properties for Monge transport density and for solutions of some shape optimization problem (2002, result): “As seen in [1],[17] this measure µ is also related to an ODE version of the optimal transport problem introduced by Brenier and explained in [ 2 ] (see also [1]).”
Convexity properties of Dirichlet integrals and Picone-type inequalities (2014, background): “This simple remark is the crucial ingredient of the so-called Benamou-Brenier formula for the 2−Wasserstein distance (see [4, 9]).”
Displacement interpolation using Lagrangian mass transport (2011, methodology): “In the special case where the cost function is quadratic and the space is Euclidean, the Kantorovich formulation can be solved with linear partial differential equations [Benamou and Brenier 2000; Haker et al. 2004; Rehman et al. 2009; Zhu et al. 2007].”
Optimal Mass Transport: Signal processing and machine-learning applications (2017, methodology): “Utilizing the Riemmanian manifold structure of ( ) P X together with the inner product presented in (9), the 2-Wasserstein metric can be reformulated into finding the minimizer of the following action among all curves in ( ) P X connecting I0 and I1 [5],”
On quotients of spaces with Ricci curvature bounded below (2017, methodology): “Using the Markov kernel, Maas [54] and Mielke [55] defined a non-local metric on the space of probability measures which imitates the Benamou-Brenier characterization of the Wasserstein metric in R [14].”
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy (2012, methodology): “Using the same symbol C(ρ0, ρ1) to denote the set of couplings between ρ0 and ρ1 in this new sense, Benamou and Brenier show that W(ρ0, ρ1) is given by W2(ρ0, ρ1) = inf {ρ,v}∈C(ρ0,ρ1) 1 2 ∫ 1 0 ∫ Rn |v(x, t)|2ρ(x, t) dxdt .”
Improving and generalizing flow-based generative models with minibatch optimal transport (2023, positive): “The Diffusion Schr¨odinger Bridge seems to outperform SB-CFM early in the trajectory, however fails to fit the bridge after many integration steps.”
Statistical Aspects of Wasserstein Distances (2018, background): “Benamou & Brenier (2000) exploit this equivalence and develop a numerical scheme to approximate the entire geodesic.”
Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures (2015, methodology): “Its dynamic characterization in terms of the continuity equation [7] and its dual formulation in terms of the Hopf-Lax formula and the corresponding (sub-)solutions of the Hamilton-Jacobi equation [34] lie at the core of the applications to gradient flows and partial differential equations of diffus”
Optimal Mass Transport for Registration and Warping (2004, methodology): “For example, methods have been proposed based on linear programming [29], and on Lagrangian mechanics closely related to ideas from the study of fluid dynamics [5].”
A survey of the Schr\"odinger problem and some of its connections with optimal transport (2013, background): “It is a remarkable fact that W2(μ0, μ1) also admits the following Benamou-Brenier representation [1]:”
Reconstruction of the early Universe as a convex optimization problem (2003, methodology): “In principle it should be possible to adapt to the Euler–Poisson reconstruction the method of the augmented Lagrangian which has been applied to the two-dimensional Monge–Ampère equation (Benamou & Brenier 2000).”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, methodology): “Both the convexity and this last expression will be useful for numerical methods (as it was first done in [11]).”