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Universität Augsburg
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Universität Augsburg
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Virginie Ehrlacher
CERMICS - Ecole des Ponts ParisTech
spricht am
Donnerstag, 16. Januar 2020
um
17:15 Uhr
im
TUM, Boltzmann-3, Garching, Raum 03.08.011, Etage 3
über das Thema:
| Abstract: |
| The aim of this talk is to present some recent results on a relaxation of multi-marginal Kantorovich optimal transport problems with a view to the design of numerical schemes to approximate the exact optimal transport problem. More precisely, the approximate problem considered in this talk consists in relaxing the marginal constraints into a finite number of moment constraints, while the state space remains unchanged (typically a subset of R^d for some positive integer d). Using Tchakhaloff’s theorem, it is possible to prove the existence of minimizers of this relaxed problem and characterize them as discrete measures charging a number of points which scales at most linearly with the number of marginals in the problem. In the particular case of a symmetric multi-marginal problem, like the Coulomb cost optimal transport problem which is the semi-classical limit of the Lévy-Lieb functional [1], the number of points charged by minimizers is independent of the number of electrons, thus avoiding the curse of dimensionality. This result is strongly linked to the work [2] and opens the way to the design of new numerical schemes exploiting the structure of these minimizers. Some preliminary numerical results exploiting this structure will be presented. [1] COTAR, Codina, FRIESECKE, Gero, et KLÜPPELBERG, Claudia. Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg–Kohn functional. Archive for Rational Mechanics and Analysis, 2018, vol. 228, no 3,, p.891-922. [2] FRIESECKE, Gero et VÖGLER, Daniela. Breaking the curse of dimension in multi-marginal kantorovich optimal transport on finite state spaces. SIAM Journal on Mathematical Analysis, 2018, vol. 50, no 4, p. 3996-4019. |
| Hierzu ergeht herzliche Einladung. |