Congrès d'Analyse Numérique pour les Jeunes - 2020

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

4 déc. 2020, 15:00

30m

Zoom (Virtuel)

Session parallèle 11

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: $d{X}_t^{ϵ}=\frac{\sigma (X_t^{ϵ})m_t^{ϵ}}{ϵ}dt$, where $d{m}_t^{ϵ}=-\frac{m_t^{ϵ}}{{ϵ}^2}dt+\frac{1}{ϵ}d{\beta }_t$. The solution $X^{ϵ}$ then converges in distribution when $ϵ\to 0$ to the solution diffusion equation $d{X}_t=\sigma (X_t)\circ d{W}_t$, 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 ($d{X}_t=\sigma (X_t)d{W}_t$). We propose an Asymptotic Preserving scheme, in the sense that the scheme converges when $ϵ\to 0$, 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 ${\partial }_t{f}^{ϵ}+\frac{1}{ϵ}v\cdot {\mathrm{\nabla }}_x{f}^{ϵ}=\frac{1}{{ϵ}^2}L{f}^{ϵ}+\frac{1}{ϵ}{m}^{ϵ}{f}^{ϵ}$, which also converge to a diffusion equation ${\partial }_t\rho =div(K\rho )+\rho \circ QdW$, 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