We consider the wellknown LiebLiniger (LL) model for
We study the convergence rate of a continuoustime simulated annealing process
 Award ID(s):
 2113779
 Publication Date:
 NSFPAR ID:
 10336723
 Journal Name:
 Numerical Algebra, Control and Optimization
 Volume:
 0
 Issue:
 0
 Page Range or eLocationID:
 0
 ISSN:
 21553289
 Sponsoring Org:
 National Science Foundation
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bosons interacting pairwise on the line via the\begin{document}$ N $\end{document} potential in the meanfield scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the timedependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the onedimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [\begin{document}$ \delta $\end{document} 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65 ,66 ,67 ] and Knowles and Pickl [44 ]. To overcome difficulties stemming from the singularity of the potential, we introduce a new shortrange approximation argument that exploits the Hölder continuity of the\begin{document}$ \delta $\end{document} body wave function in a single particle variable. By further exploiting the\begin{document}$ N $\end{document} subcritical wellposedness theory for the 1D cubic NLS, we can prove meanfield convergence when the limiting solution to the NLS has finitemore »\begin{document}$ L^2 $\end{document} 
Consider the linear transport equation in 1D under an external confining potential
:\begin{document}$ \Phi $\end{document} For
(with\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} small), we prove phase mixing and quantitative decay estimates for\begin{document}$ \varepsilon >0 $\end{document} , with an inverse polynomial decay rate\begin{document}$ {\partial}_t \varphi : =  \Delta^{1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document} . In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in\begin{document}$ O({\langle} t{\rangle}^{2}) $\end{document} D under the external potential\begin{document}$ 1 $\end{document} .\begin{document}$ \Phi $\end{document} 
The disparity in the impact of COVID19 on minority populations in the United States has been well established in the available data on deaths, case counts, and adverse outcomes. However, critical metrics used by public health officials and epidemiologists, such as a time dependent viral reproductive number (
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conservative map\begin{document}$ C^d $\end{document} of the\begin{document}$ f $\end{document} dimensional unit ball\begin{document}$ d $\end{document} ,\begin{document}$ {\mathbb B}^d $\end{document} , can be realized by renormalized iteration of a\begin{document}$ d\geq 2 $\end{document} perturbation of identity: there exists a conservative diffeomorphism of\begin{document}$ C^d $\end{document} , arbitrarily close to identity in the\begin{document}$ {\mathbb B}^d $\end{document} topology, that has a periodic disc on which the return dynamics after a\begin{document}$ C^d $\end{document} change of coordinates is exactly\begin{document}$ C^d $\end{document} .\begin{document}$ f $\end{document} 
Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large
player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into\begin{document}$ N $\end{document} suboptimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an\begin{document}$ N $\end{document} Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.\begin{document}$ \epsilon $\end{document}