3–4 déc. 2020
Virtuel
Fuseau horaire Europe/Paris

On Asymptotic Preserving schemes for some SDEs and SPDEs in the diffusion approximation regime.

4 déc. 2020, 15:00
30m
Zoom (Virtuel)

Zoom

Virtuel

Salle 1 : https://zoom.us/j/94929969299 Salle 2 : https://zoom.us/j/98740649245 Salle 3 : https://zoom.us/j/99534523679

Orateur

M. SHMUEL RAKOTONIRINA--RICQUEBOURG (Institut Camille Jordan, UCB Lyon 1)

Description

We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations (SDE). We focus on an example in the so-called diffusion approximation regime: $dX^\epsilon_t = \frac{\sigma(X^\epsilon_t) m^\epsilon_t}{\epsilon} dt$, where $dm^\epsilon_t = -\frac{m^\epsilon_t}{\epsilon^2} dt + \frac{1}{\epsilon} d\beta_t$. The solution $X^\epsilon$ then converges in distribution when $\epsilon \to 0$ to the solution diffusion equation $dX_t = \sigma(X_t) \circ dW_t$, with a Stratonovitch interpretation of the noise $W$. The natural schemes fail to capture the correct limiting equation, as they give a limit scheme consistent with an Itô interpretation of the noise ($dX_t = \sigma(X_t)dW_t$). We propose an Asymptotic Preserving scheme, in the sense that the scheme converges when $\epsilon \to 0$, and that the limit scheme is consistent with the limiting equation with the correct interpretation of the noise. We also present a kinetic stochastic PDE $\partial_t f^\epsilon + \frac{1}{\epsilon} v \cdot \nabla_x f^\epsilon = \frac{1}{\epsilon^2} L f^\epsilon + \frac{1}{\epsilon} m^\epsilon f^\epsilon$, which also converge to a diffusion equation $\partial_t \rho = div(K \rho) + \rho \circ Q dW$, and some ideas on how to construct AP schemes for this SPDE.

Preprints: https://arxiv.org/abs/2011.02341, https://arxiv.org/abs/2009.10406

Auteurs principaux

M. SHMUEL RAKOTONIRINA--RICQUEBOURG (Institut Camille Jordan, UCB Lyon 1) M. Charles-Édouard Bréhier (ICJ, Université Lyon 1)

Documents de présentation

Aucun document.