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

TL;DR

This paper introduces a variance reduction technique for simulating bacterial chemotaxis models by coupling detailed internal dynamics with a simplified gradient sensing process, leading to asymptotic variance reduction in the diffusive limit.

Contribution

It presents a novel hybrid simulation scheme that reduces variance by coupling complex internal dynamics with a simplified model using shared randomness.

Findings

Variance reduction is asymptotic in the diffusive limit.

Coupling reduces simulation variance effectively.

Hybrid scheme improves computational efficiency.

Abstract

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variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations.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.