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Session III.3 - Computational Optimal Transport

Monday, June 19, 18:00 ~ 18:30

Sparsity results for moment-constrained approximation of the Lieb functional

Virginie Ehrlacher

Ecole des Ponts ParisTech & INRIA, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The aim of this talk is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations for molecules. The Lieb functional was actualy shown tby Lieb o be a convexification of the so-called Lévy-Lieb functional. Given an electronic density for a system of N electrons, which may be seen as a probability density defined on the set R^3, the value of the Lieb functional for this density is defined as the solution of a quantum multi-marginal optimal transport problem, which reads as a minimization problem defined onto the set of trace-class operators acting on the space of electronic wavefunctions that are antisymmetric L^2 functions of R^{3N}, with partial trace equal to the prescribed electronic density. We introduce a relexation of this quantum optimal transport where the full partial trace constraint is replaced by a finite number of moment constraints on the partial trace of the set of operators. We show that, provided that the electronic density decays to 0 at infinity fast enough, there exists sparse minimizers to the moment-constrained approximation of the Lieb (MCAL) functional that read as operators with rank at most equal to the number of moment constraints. We also prove under appropriate assumptions on the set of moment functions that the value of the MCAL functional converges to the value of the exact Lieb functional as the number of moments go to infinity.

Joint work with Luca Nenna (Laboratoire de Mathématiques d'Orsay, France).

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