Institutskolloquium Sommersemester 2020

Das Institutskolloquium findet während der Vorlesungszeit an jedem Donnerstag um 17:15 Uhr im Raum 05-432 (Hilbertraum) statt. Ab 16:45 Uhr gibt es Kaffee und Kuchen.
Aufgrund der derzeit geltenden Distanzmaßnahmen findet das Institutskolloquium in diesem Semester nur in eingeschränktem Umfang und als online-Veranstaltung statt.

Programm

02.07.2020 17 Uhr c.t. Dr. Virginie Ehrlacher (École des Ponts)
Moment constrained optimal transport problem: application to quantum chemistry

Online event, access via BigBluButton: https://bbb1.physik.uni-mainz.de/b/kol-6c9-j7d
(preferably using a firefox or chrome browser)

 

Abstracts:

02.07.2020 17 Uhr c.t. Dr. Virginie Ehrlacher (École des Ponts): Moment constrained optimal transport problem: application to quantum chemistry (joint work with A. Alfonsi, R. Coyaud and D. Lombardi)

Online event, access via BigBluButton: https://bbb1.physik.uni-mainz.de/b/kol-6c9-j7d
(preferably using a firefox or chrome browser)

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.
After a general introduction to Density Functional Theory and optimal transport problems, I will explain in this talk how the semi-classical limit of the so-called Lévy-Lieb functional, which is a central quantity in electronic structure computations,
happens to be a symmetric multimarginal optimal tranport problem with Coulomb cost [1]. I will then present one particular numerical method which can be used for the resolution of such problem and avoids the curse of dimensionality. More precisely, the approximate problem considered 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 arising in quantum chemistry applications, 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.

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