Lecture Notes on Optimal Transport Problems

By Luigi Ambrosio. 2003

432 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

Inference for empirical Wasserstein distances on finite spaces (2016, background): “, 2000) or optimal transport distance in optimization (Ambrosio, 2003), is one of the most fundamental metrics on the space of probability measures. Besides its prominence in probability (e.g. Dobrushin (1970); Gray (1988)) and finance (e.g. Rachev and Rüschendorf (1998)) it has deep connections to ”
On the sufficiency of c-cyclical monotonicity for optimality of transport plans (2008, background): “Concerning the opposite implication, namely that a transport plan concentrated in a c-cyclical monotone set is optimal, it is known since a long time if the cost function c is continuous and the ambient space is a compact set X = Y ⊆ Rn , see for instance [1,8,15]; recently, this result has been ext”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, methodology): “For subsequent developments in this direction, see works of Ambrosio, CorderoErausquin, Carrillo, Figalli, Gigli, Lott, Maggi, McCann, Nazaret, Otto, Pratelli, von Renesse, Schmuckenschläger, Sturm, Topping and Villani in [95] [110] [30] [31] [32] [5] [24] [116] [127] [88] [87] [99] [52].”
Optimal Mass Transport for Registration and Warping (2004, background): “An excellent summary of the recent research on this topic may be also found [1].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “In spite of the fact that this latter cost does not satisfy any of the conditions (C0)-(C3), still existence of optimal transport maps can be proved (see for instance [6] for an account of the theory) and some regularity results have been obtained in [62].”
Interior regularity of optimal transport paths (2004, background): “Introduction In a previous paper [12], the author considered a phenomena of mass transport problems, for which the actual transport cost is given by the actual transport path, rather than by transportmaps as inMonge’s problems [7,10,8,3,6,2].”
An optimal transport approach for seismic tomography: application to 3D full waveform inversion (2016, background): “In addition, when (14) admits a solution T , the measure π = (I, T )#µ is a solution of the relaxed problem (13) [Ambrosio, 2003, Pratelli, 2007].”
Optimal transport from Lebesgue to Poisson (2010, background): “The claim basically relies on the following lemma which is a slight modification (and extension) of a result in [Amb03].”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “This is illustrated by the following simple example, which also indicates why new hypotheses are required to ensure uniqueness. principle have been examined extensively in the context of dynamical problems since the work of Otto [65] and Jordan, Kinderlehrer and Otto [50], see also e.g. Agueh [2], A”
OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS (2003, neutral): “Communications in Contemporary Mathematics Vol. 5, No. 2 (2003) 251–279 c© World Scientific Publishing Company OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS QINGLAN XIA Rice University, Mathematics, Houston, TX,77005, USA qlxia@rice.edu Received 4 December 2001 Revised 1 July 2002 In transport…”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, methodology): “This has been observed by Ambrosio and fixed in [8, 10].”
Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions (2007, neutral): “See [1, 27, 52] for more on optimal transport maps.”