模拟细菌趋化性的个体模型与渐近方差缩减
Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction
AI总结：
本文通过耦合细菌趋化性的个体模型与直接梯度感知的简化过程，提出一种渐近方差缩减的混合模拟方案。
摘要：
我们讨论了具有内部动力学的细菌趋化性个体模型的方差缩减模拟。方差缩减是通过将该模型与一个更简单的过程耦合实现的，其中内部动力学被替换为对趋化剂浓度的直接梯度感知。在配套论文\cite{limits}中，我们使用路径概率技术严格证明，在扩散渐近下，两个过程收敛于相同的平流-扩散过程。本文中，通过在两模拟中使用相同的随机数集，实现了两个模型模拟的个体细菌路径的直接耦合。该耦合用于构建具有缩减方差的混合方案。我们首先计算直接梯度感知模型的动力学密度描述的确定性解；然后通过耦合的个体模拟评估内部动力学引起的偏差。我们证明得到的方差缩减是\emph{渐近的}，即在扩散渐近下，两个过程之间的差异的方差根据小参数消失。
英文摘要：
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variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations.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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