Section: New Results

Analysis of the Monte-Carlo error in a hybrid semi-Lagrangian scheme

In [17] we consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. The goal of this paper is twofold. First we give rigorous convergence estimates for our algorithm: In a simple setting, we show that under an anti-CFL condition on the time-step $\delta t$ and on the mesh size $\delta x$ and for a reasonably large number of independent realizations $N$, we control the Monte-Carlo error by a term of order $𝒪\left(\sqrt{\delta t/N}\right)$. Then, we show various applications of the numerical method in very general situations (nonlinear, different boundary conditions, higher dimension) and numerical examples showing that the theoretical bound obtained in the simple case seems to persist in more complex situations.