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Étude d'approximations de problèmes de transport optimal et application à la physique

Rafaël Coyaud 1, 2 
2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
Abstract : Optimal Transport (OT) problems arise in numerous applications. Numerical approximation of these problems is a practical challenging issue. We investigate a relaxation of OT problems when marginal constraints are replaced by some moment constraints (MCOT problem), and show the convergence of the latter towards the former. Using Tchakaloff's theorem, we show that the MCOT problem is achieved by a finite discrete measure. For multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which allows to bypass the curse of dimension. This method is also relevant for Martingale OT problems. In some fundamental cases, we get rates of convergence in [dollar]O(1/n)[dollar] or [dollar]O(1/n^2)[dollar] where [dollar]n[dollar] is the number of moments, which illustrates the role of the moment functions.We design a numerical method, built upon constrained overdamped Langevin processes, to solve MCOT problems; and proved that any local minimizer to the MCOT problem is a global one. We provide numerical examples for large symmetrical multimarginal MCOT problems.We extend a method (E. Canc`es and L.R. Scott, SIAM J. Math. Anal., 50, 2018, 381--410) to compute more terms in the asymptotic expansion of the van der Waals attraction between two hydrogen atoms. These terms are obtained by solving a set of modified Slater--Kirkwood PDE's. The accuracy of the method is demonstrated by numerical simulations and comparison with other methods from the literature. We also show that the scattering states of the hydrogen atom (the ones associated with the continuous spectrum of the Hamiltonian) have a major contribution to the C[dollar]_6[dollar] coefficient of the van der Waals expansion.
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Rafaël Coyaud. Étude d'approximations de problèmes de transport optimal et application à la physique. Mathématiques générales [math.GM]. Université Paris-Est, 2021. Français. ⟨NNT : 2021PESC1103⟩. ⟨tel-03529782⟩

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