Section: New Results

Coupling methods and variance reduction

Participant : Mathias Rousset.

Recently, M. Rousset has initiated a research topic on variance reduction techniques (called ”asymptotic”) for the simulation of stochastic models of particles. The point is to use a macroscopic (or model reduced) equation as a control variate; or in other words, to use the information of a macroscopic description to decrease the statistical error of the simulated microscopic evolution.

A first step in this program has been achieved for a microscopic model describing the individual motion of bacteriae with a Markovian velocity-jump process. The macroscopic equation is an advection-diffusion equation called the chemotaxis equation. In [30] , the pobabilistic derivation of the chemotaxis equation from the individual motion of bacteriae have been carried out in a rigorous way. In [31] , a numerical method simulating the individual evolution of bacteriae with ”asymptotic” variance reduction have been proposed.

Motivated by the asymptotic variance reduction of DSMC methods (particle Monte-Carlo methods simulating low density fluids modeled by kinetic equations), the work in [50] , M. Rousset considers space homogenous Boltzmann kinetic equations in dimension d with Maxwell collisions (and without Grad's cut-off). An explicit Markov coupling of the associated conservative (Nanbu) stochastic N-particle system is constructed, using plain parallel coupling of isotropic random walks on the sphere of two-body collisional directions. The resulting coupling is almost surely decreasing, and the $L_2$-coupling creation is computed explicitly. Some quasi-contractive and uniform in N coupling / coupling creation inequalities are then proved, relying on 2 + $\alpha$-moments ($\alpha$ > 0) of velocity distributions; upon N-uniform propagation of moments of the particle system, it yields a N-scalable $\alpha$-power law trend to equilibrium. The latter are based on an original sharp inequality, which bounds from above the coupling distance of two centered and normalized random variables $(U,V)\in {ℝ}^d$, with the average square parallelogram area spanned by $(U-U_{*},V-V_{*})$, $(U_{*},V_{*})$ denoting an independent copy. Two counter-examples proving the necessity of the dependance on > 2-moments and the impossibility of strict contractivity are provided. The paper, (mostly) self-contained, does not require any propagation of chaos property and uses only elementary tools.