Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

By Luis Caffarelli, Mikhail Feldman, Robert J. McCann. 2001

210 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3ebf685a07eb2dbc595b0e54aa5e1b622a16f492

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Regularity properties for Monge transport density and for solutions of some shape optimization problem (2002, result): “when d in (1.1) is replaced by a strictly convex,positive function of d) and much less in the case of linear costs,as in this paper and in [ 6 ].”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “After that, many results reduced the assumptions on the supports of μ0, μ1, see [6] and [26].”
Optimal transport maps in Monge-Kantorovich problem (2003, methodology): “The following result [3] is a slight improvement of [12], where existence of an optimal transport map was estabilished but not the stability property.”
Optimal transportation, topology and uniqueness (2010, background): “Still, 50 more years would elapse before the optimal volume-preserving map between two arbitrary domains sought by Monge was constructed for the Euclidean distance c(x, y) = |x − y| by Ambrosio [4], Caffarelli, Feldman, McCann [22], and Trudinger, Wang [99].”
Regularity of Potential Functions of the Optimal Transportation Problem (2005, background): “The original cost function of Monge was proportional to the distance moved and surprisingly this problem was only recently completely solved [9,29], see also [2].”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “Existence of solutions is tricky to establish: the first ‘proof’, due to Sudakov [128], relied on an unsubstantiated claim which turned out to be correct only in the plane M± = R(2) [11]; higher dimensional arguments were given increasing generality by Evans-Gangbo [44], and then Ambrosio [4], Caffa”
Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes (2004, unknown): “…problem for which the Euler-Lagrange equation is the p - Laplacian PDE under the assumption that P 0 and P 1 have disjoint compact supports, and in [6] and [24] they studied (1.2) by the “ q ↓ 1” limit of (1.2) with L ( u ) = | u | q under the assumption that P 0 and P 1 have compact supports (see…”
Discriminator optimal transport (2019, methodology): “However, there is a known method [26] to find a solution based on relaxing the cost to strictly convex cost ||x − y|| 2 with ε > 0.”
Lecture Notes on Optimal Transport Problems (2003, methodology): “For stricly convex norms, the first fully rigorous proofs of the existence of an optimal transport map under the only assumption that f0 15 ] and [43].”
Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem (2003, methodology): “We will also use the following constructions from [8].”
A User’s Guide to Optimal Transport (2013, unknown): “McCann [24].”
Optimal transportation on non-compact manifolds (2007, background): “However for the case where the cost is a distance, like in (1), one should cite at least the work of Sudakov [39], Evans-Gangbo [19], Feldman-McCann [23], Caffarelli-Feldman-McCann [11], Ambrosio-Pratelli [5], and Bernard-Buffoni [8].”
Monge's transport problem on a Riemannian manifold (2001, methodology): “The proof of this estimate is a main technical difficulty distinguishing the present Riemannian problem from the Euclidean case studied in [3] and [19].”
On the design of a reflector antenna II (2004, background): “For Monge’s cost function the existence of optimal mappings was not proved until recently in [2,8,19].”
On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation (2007, background): “the Lebesgue measure (see [1], or also [4,3,13, 2,11] or [10] for general references).”