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

Mathias Rousset 1, 2, 3 Giovanni Samaey 4
1 MICMAC - Methods and engineering of multiscale computing from atom to continuum
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
3 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
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variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations.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.
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Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013, 23 (12), pp.2155 - 2191. 〈10.1142/S0218202513500292〉
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Mathias Rousset, Giovanni Samaey. Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013, 23 (12), pp.2155 - 2191. 〈10.1142/S0218202513500292〉. 〈hal-00643324〉

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