In this talk, I will present a recent work where we study a new multilevel method to sample Gibbs distribution pi over Rd, based on (overdamped) Langevin approximation. This method both relies on a multilevel occupation measure, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with geometrically decreasing step size. We first show that the parameters of the numerical strategy can be chosen in such a way that a epsilon-approximation of the target (in a L2-sense) can be done with a cost proportional to epsilon - 2, i.e. proportional to a Monte-Carlo method without bias.
In a second part, we investigate the effect of the dimension d of the problem and optimize the choice of the parameters of the multilevel strategy in terms of d. In the uniformly strongly convex setting, this leads to a complexity proportional to d epsilon - 2log^3(d epsilon - 2) and to depsilon - 2 under additional assumptions.