A stochastic particle method for the McKean-Vlasov and the Burgers equation

Abstract:

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in $[0,T]\times \mathbb {R}$ with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure $\mu _t$, the solution to the McKean-Vlasov equation. The simulation of the stochastic system with $N$ particles provides a discrete measure which approximates $\mu _{k\delta t}$ for each time $k\delta t$ (where $\delta t$ is a discretization step of the time interval $[0,T]$). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of $\mu _{k\delta t}$. We show that the convergence rate is ${\mathcal O}\left (1/\sqrt {N}+\sqrt {\delta t}\right )$ for the approximation in $L^1(\Omega \times \mathbb {R})$ of the cumulative distribution function at time $T$, and of order ${\mathcal O}\left (\varepsilon ^2 + \frac {1}{\varepsilon } \left (\frac {1}{\sqrt {N}}+ \sqrt {\delta t}\right )\right )$ for the approximation in $L^1(\Omega \times \mathbb {R})$ of the density at time $T$ ($\Omega$ is the underlying probability space, $\varepsilon$ is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate ${\mathcal O}\left (1/\sqrt {N}+\sqrt {\delta t}\right )$. This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.