Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law

Abstract

In this paper, we analyse the rate of convergence of a system of N interacting particles with meanfield rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov[18] to check trajectorial propagation of chaos with optimal rate N −1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy [3] to check convergence in L 1 (R) with rate O 1 √ N + h of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in O 1 N + h . We provide numerical results which confirm our theoretical estimates.

Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

“…In Section 2, we show that when µ is compactly supported, the order of convergence of e N (µ, ρ) to 0 is not smaller than 1 ρ and that when the quantile function F −1 is discontinuous, it is not greater than 1 ρ . Note that, in these two situations, the corresponding threshold is 1 2ρ for E 1/ρ W ρ ρ (µ N , µ) .…”

“…In Section 2, we show that when µ is compactly supported, the order of convergence of e N (µ, ρ) to 0 is not smaller than 1 ρ and that when the quantile function F −1 is discontinuous, it is not greater than 1 ρ . Note that, in these two situations, the corresponding threshold is 1 2ρ for E 1/ρ W ρ ρ (µ N , µ) . Then we state our first theorem which bounds lim inf N →+∞ N e N (µ, ρ) from below by some value involving the density f of the absolutely continuous with respect to the Lebesgue measure part of µ and ensures that N e N (µ, ρ) goes to this value as N → +∞ when the density f is dx a.e.…”

“…In general dimension, estimations of E 1/ρ W ρ ρ (µ N , µ) and concentration inequalities for W ρ ρ (µ N , µ) are given in [7]. In the random case, the largest possible order of convergence (still apart from the case when µ is a Dirac mass) is α = 1 2 , which matches the rate of convergence in the strong law of large numbers given by the central limit theorem under square integrability.…”

“…We also exhibit non-compactly supported probability measures µ such that, for ρ > 1, lim N →+∞ N 1 ρ e N (µ, ρ) = 0. Nevertheless, for (N α e N (µ, 1)) N ≥1 to be bounded for α > 1 ρ , µ has to be compactly supported. We last give a necessary condition for N 1 ρ e N (µ, 1)…”

“…The motivation is the approximation of the probability measure µ by finitely supported probability measures. An example of application is provided by the optimal initialization of systems of particles with mean-field interaction [9,1], where, to preserve the meanfield feature, it is important to get N points with equal weight 1 N (of course, nothing prevents several of these points to be equal). The Hoeffding-Fréchet or comonotone coupling between two probability measures ν and η on the real line is optimal for W ρ so that:…”