Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction

We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper \cite{limits}, we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is \emph{asymptotic}, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter.